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321
TIFR ECE 2018 | Question: 1
Consider a discrete-time system which in response to input sequence $x[n] \;( n$ integer) outputs the sequence $y[n]$ such that \[y[n]=\left\{\begin{array}{ll} 0, & n=-1,-2,-3, \ldots, \\ \frac{1}{2 ... describes the system? Linear, time-invariant Linear, time-variant Non-linear, time-invariant Non-linear, time-variant Cannot be determined from the information given
Consider a discrete-time system which in response to input sequence $x[n] \;( n$ integer) outputs the sequence $y[n]$ such that\[y[n]=\left\{\begin{array}{ll}0, & n=-1,-2...
admin
46.4k
points
98
views
admin
asked
Nov 29, 2022
Others
tifrece2018
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1
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322
TIFR ECE 2018 | Question: 2
A hotel has $n$ rooms numbered $1,2, \ldots, n$. For each room there is one spare key labeled with the room number. The hotel manager keeps all the spare keys in a box. Her mischievous son got hold of the box and permuted the labels uniformly at random. What is the ... Use linearity of expectation] $1$ $\frac{n-1}{n}$ $\frac{n}{n-1}$ $\frac{n}{2}$ None of the above
A hotel has $n$ rooms numbered $1,2, \ldots, n$. For each room there is one spare key labeled with the room number. The hotel manager keeps all the spare keys in a box. H...
admin
46.4k
points
90
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
expectation
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1
votes
0
answers
323
TIFR ECE 2018 | Question: 3
Let $\lim _{n \rightarrow \infty} f(n)=\infty$ and $\lim _{n \rightarrow \infty} g(n)=\infty$. Then which of the following is necessarily $\text{TRUE.}$ $\lim _{n \rightarrow \infty}|f(n)-g(n)|=\infty$ $\lim _{n \rightarrow \infty}|f(n)-g(n)|=0$ $\lim _{n \rightarrow \infty}|f(n) / g(n)|=\infty$ $\lim _{n \rightarrow \infty}|f(n) / g(n)|=1$ None of the above
Let $\lim _{n \rightarrow \infty} f(n)=\infty$ and $\lim _{n \rightarrow \infty} g(n)=\infty$. Then which of the following is necessarily $\text{TRUE.}$$\lim _{n \rightar...
admin
46.4k
points
124
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
limits
+
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1
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0
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324
TIFR ECE 2018 | Question: 4
Consider \[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\] What can we say about $\lim _{x \rightarrow \infty} f(x)$ ? The function $f(x)$ does not have a limit as $x \rightarrow \infty$ ... $\lim _{x \rightarrow \infty} f(x)=e^{1 / 2}$ $\lim _{x \rightarrow \infty} f(x)=0$ $\lim _{x \rightarrow \infty} f(x)=\infty$
Consider\[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\]What can we say about $\lim _{x \rightarrow \infty} f(x)$ ?The function $f(x)$ ...
admin
46.4k
points
82
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
limits
+
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1
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0
answers
325
TIFR ECE 2018 | Question: 5
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such that they solve the following optimization problem \[d=\min _{a_{1}, a_{2} \in \mathbb{R}}\left\ ... $\left\|\vec{v}_{*}\right\|^{2}-\|\vec{u}\|^{2}$ None of the above
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such ...
admin
46.4k
points
93
views
admin
asked
Nov 29, 2022
Vector Analysis
tifrece2018
vector-analysis
vector-in-planes
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1
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0
answers
326
TIFR ECE 2018 | Question: 6
Consider the system shown below. If $K>0$, which of the following describes the system? Stable, causal Stable, non-causal Unstable, non-causal Unstable, causal Cannot be determined from the information given
Consider the system shown below.If $K>0$, which of the following describes the system?Stable, causalStable, non-causalUnstable, non-causalUnstable, causalCannot be determ...
admin
46.4k
points
106
views
admin
asked
Nov 29, 2022
Others
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1
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327
TIFR ECE 2018 | Question: 7
Let $X_{1}, X_{2}$ and $X_{3}$ be independent random variables with uniform distribution over $[0, \theta]$. Consider the following statements. $E\left[\max \left\{X_{1}, X_{2}, X_{3}\right\}\right]=\frac{3}{4} \theta$ ... $\text{(i)}$ Only $\text{(ii)}$ Only $\text{(iii)}$ Only $\text{(iv)}$ All of $\text{(i) - (iv)}$
Let $X_{1}, X_{2}$ and $X_{3}$ be independent random variables with uniform distribution over $[0, \theta]$. Consider the following statements.$E\left[\max \left\{X_{1}, ...
admin
46.4k
points
93
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
uniform-distribution
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1
votes
0
answers
328
TIFR ECE 2018 | Question: 8
Let $A$ be an $n \times n$ real matrix for which two distinct non-zero $n$-dimensional real column vectors $v_{1}, v_{2}$ satisfy the relation $A v_{1}=A v_{2}$. Consider the following statements. At least one eigenvalue of $A$ is zero. $A$ ... $\text{(i)}$ Only $\text{(ii)}$ Only $\text{(iii)}$ Only $\text{(iv)}$ All of $\text{(i) - (iv)}$
Let $A$ be an $n \times n$ real matrix for which two distinct non-zero $n$-dimensional real column vectors $v_{1}, v_{2}$ satisfy the relation $A v_{1}=A v_{2}$. Consider...
admin
46.4k
points
92
views
admin
asked
Nov 29, 2022
Linear Algebra
tifrece2018
linear-algebra
matrices
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1
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0
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329
TIFR ECE 2018 | Question: 9
Let $X$ and $Y$ be two independent and identically distributed binary random variables that take values $\{-1,+1\}$ each with probability $1 / 2$. Let $Z_{1}=\max (X, Y)$, and $Z_{2}=\min (X, Y)$. Consider the following statements. $Z_{1}$ and $Z_{2}$ are uncorrelated ... $\text{(iii)}$ Both $\text{(i) and (ii), but not (iii)}$ All of $\text{(i), (ii) and (iii)}$
Let $X$ and $Y$ be two independent and identically distributed binary random variables that take values $\{-1,+1\}$ each with probability $1 / 2$. Let $Z_{1}=\max (X, Y)$...
admin
46.4k
points
116
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
random-variable
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1
votes
0
answers
330
TIFR ECE 2018 | Question: 10
Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice each of which takes six values $1,2,3,4,5,6$ with equal probability. What is the conditional expectation \[E\left[X_{1} \mid \max \left(X_{1}, X_{2}\right)=5\right]\] $3$ $4$ $35 / 9$ $5 / 2$ $15 / 4$
Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice each of which takes six values $1,2,3,4,5,6$ with equal probability. What is ...
admin
46.4k
points
100
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
331
TIFR ECE 2018 | Question: 11
Assume the following well known result: If a coin is flipped independently many times and its probability of heads $(H)$ is $p \in(0,1)$ and probability of tails $(T)$ is $(1-p)$, then the expected number of coin flips till the first time a heads is observed is $1 / p$. What is the ... $\frac{1}{1-(1-p)^{2}}(4+1 / p)$ $\frac{1}{p}+\frac{1}{1-p}$
Assume the following well known result: If a coin is flipped independently many times and its probability of heads $(H)$ is $p \in(0,1)$ and probability of tails $(T)$ is...
admin
46.4k
points
93
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
expectation
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1
votes
0
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332
TIFR ECE 2018 | Question: 12
Suppose that Amitabh Bachchan has ten coins in his pocket. $3$ coins have tails on both sides. $4$ coins have heads on both sides. $3$ coins have heads on one side and tails on the other and both the outcomes are equally likely when that coin is flipped. In a bet with Dharmendra ... that the other side of this coin is heads? $1 / 2$ $3 / 10$ $1 / 4$ $0.3$ $1 / 3$
Suppose that Amitabh Bachchan has ten coins in his pocket. $3$ coins have tails on both sides. $4$ coins have heads on both sides. $3$ coins have heads on one side and ta...
admin
46.4k
points
111
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
conditional-probability
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1
votes
0
answers
333
TIFR ECE 2018 | Question: 13
Consider five distinct binary vectors $X_{1}, \ldots, X_{5}$ each of length $10$. Let \[d_{i j}=\sum_{k=1}^{10}\left(X_{i k} \text { XOR } X_{j k}\right),\] (i.e., $d_{i j}$ is the number of coordinates where $X_{i}$ ... to $X_{5}$, and argue about the result noting that there are five binary vectors.] $d=10$ $d=9$ $d=8$ $d<8$ Information is not sufficient
Consider five distinct binary vectors $X_{1}, \ldots, X_{5}$ each of length $10$. Let\[d_{i j}=\sum_{k=1}^{10}\left(X_{i k} \text { XOR } X_{j k}\right),\](i.e., $d_{i j}...
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46.4k
points
84
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admin
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Nov 29, 2022
Others
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334
TIFR ECE 2018 | Question: 14
Define the $\ell_{p}$ ball in two dimensions as the set of points $(x, y)$ such that $|x|^{p}+|y|^{p} \leq 1$. Which of the following is $\text{FALSE:}$ The $\ell_{2}$ ball is contained in the $\ell_{3}$ ball The $\ell_{2}$ ball is contained in ... $\ell_{2}$ ball is contained in the $\ell_{5}$ ball The $\ell_{1}$ ball is contained in the $\ell_{3}$ ball
Define the $\ell_{p}$ ball in two dimensions as the set of points $(x, y)$ such that $|x|^{p}+|y|^{p} \leq 1$. Which of the following is $\text{FALSE:}$The $\ell_{2}$ bal...
admin
46.4k
points
81
views
admin
asked
Nov 29, 2022
Others
tifrece2018
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0
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335
TIFR ECE 2018 | Question: 15
Consider real-valued continuous functions $f:[0,2] \rightarrow(-\infty, \infty)$ and let \[A=\int_{0}^{1}|f(x)| d x \quad \text { and } B=\int_{1}^{2}|f(x)| d x .\] Which of the following is $\text{TRUE}?$ There exists an $f$ so that \[A+B<\int_{0}^{2} f(x) ... such that $\int_{0}^{1} f(x) d x=3$ There does not exist an $f$ so that \[A+B \leq-\int_{0}^{2} f(x) d x\]
Consider real-valued continuous functions $f:[0,2] \rightarrow(-\infty, \infty)$ and let\[A=\int_{0}^{1}|f(x)| d x \quad \text { and } B=\int_{1}^{2}|f(x)| d x .\]Which o...
admin
46.4k
points
100
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
definite-integrals
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1
votes
0
answers
336
TIFR ECE 2017 | Question: 1
Consider a system which in response to input $x(t)$ outputs \[ y(t)=2 x(t-2)+x(2 t-1)+1 . \] Which of the following describes the system? linear, time-invariant, causal linear, time-invariant, non-causal non-linear, time-invariant, causal non-linear, time-invariant, non-causal non-linear, time-variant
Consider a system which in response to input $x(t)$ outputs\[y(t)=2 x(t-2)+x(2 t-1)+1 .\]Which of the following describes the system?linear, time-invariant, causallinear,...
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46.4k
points
67
views
admin
asked
Nov 29, 2022
Others
tifrece2017
+
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1
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0
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337
TIFR ECE 2017 | Question: 2
Suppose a $1 \mu \mathrm{H}$ inductor and a $1 \Omega$ resistor are connected in series to a $1 \mathrm{ V}$ battery. What happens to the current in the circuit? The current starts at $0 \mathrm{ A}$, and gradually rises to $1 \mathrm{ A}$ The current ... $1 \mathrm{ A}$ The current oscillates over time between $1 \mathrm{ A}$ and $-1 \mathrm{ A}$
Suppose a $1 \mu \mathrm{H}$ inductor and a $1 \Omega$ resistor are connected in series to a $1 \mathrm{ V}$ battery. What happens to the current in the circuit?The curre...
admin
46.4k
points
87
views
admin
asked
Nov 29, 2022
Others
tifrece2017
+
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338
TIFR ECE 2017 | Question: 3
What is the maximum average power that can be dissipated by a load connected to the output terminals of the following circuit with an alternating current source? $23 \mathrm{~W}$ $11.5 \mathrm{~W}$ $8.1317 \mathrm{~W}$ $2.875 \mathrm{~W}$ None of the above
What is the maximum average power that can be dissipated by a load connected to the output terminals of the following circuit with an alternating current source?$23 \math...
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46.4k
points
80
views
admin
asked
Nov 29, 2022
Others
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339
TIFR ECE 2017 | Question: 4
A Schmitt trigger circuit is a comparator circuit with a hysteresis. Consider the Schmitt trigger circuit in the figure implemented using an opamp. What are the trigger levels for this circuit? $\pm \frac{R_{1}}{R_{2}} V_{s}$ $\pm \frac{R_{2}}{R_{1}} V_{s}$ $\pm \frac{R_{1}}{R_{1}+R_{2}} V_{s}$ $\pm \frac{R_{2}}{R_{1}+R_{2}} V_{s}$ None of the above
A Schmitt trigger circuit is a comparator circuit with a hysteresis. Consider the Schmitt trigger circuit in the figure implemented using an opamp. What are the trigger l...
admin
46.4k
points
101
views
admin
asked
Nov 29, 2022
Others
tifrece2017
+
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1
votes
0
answers
340
TIFR ECE 2017 | Question: 5
Consider the inequality \[ n-\frac{1}{n} \geq \sqrt{n^{2}-1}, \] where $n$ is an integer $\geq 1$. Which of the following statements is $\text{TRUE?}$ This inequality holds for all integers $n \geq 1$ ... not hold for any integer $n \geq 1$ $n-\frac{1}{n}=\sqrt{n^{2}-1}$ for infinitely many integers $n \geq 1$
Consider the inequality\[n-\frac{1}{n} \geq \sqrt{n^{2}-1},\]where $n$ is an integer $\geq 1$. Which of the following statements is $\text{TRUE?}$This inequality holds fo...
admin
46.4k
points
88
views
admin
asked
Nov 29, 2022
Quantitative Aptitude
tifrece2017
quantitative-aptitude
inequality
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–
1
votes
0
answers
341
TIFR ECE 2017 | Question: 6
Let $a, b \in\{0,1\}$. Consider the following statements where $*$ is the $\text{AND}$ operator, $\oplus$ is $\text{EXCLUSIVE-OR,}$ and ${ }^{c}$ denotes the complement function. $\max \left\{a * b, b \oplus a^{\mathrm{c}}\right\}=1$ ... $\text{(iii)}$ only $\text{(iii)}$ and $\text{(iv)}$ only $\text{(iv)}$ and $\text{(i)}$ only None of the above
Let $a, b \in\{0,1\}$. Consider the following statements where $*$ is the $\text{AND}$ operator, $\oplus$ is $\text{EXCLUSIVE-OR,}$ and ${ }^{c}$ denotes the complement f...
admin
46.4k
points
133
views
admin
asked
Nov 29, 2022
Calculus
tifrece2017
calculus
functions
+
–
1
votes
0
answers
342
TIFR ECE 2017 | Question: 7
A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a $3 \times 3$ circulant matrix. \[\left(\begin{array}{lll} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \ ... $j=\sqrt{-1}$ A vector whose $k$-th element is $\sinh \left(\frac{2 \pi k}{n}\right)$ None of the above
A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a $3 \times 3$ circulant matrix.\[\l...
admin
46.4k
points
88
views
admin
asked
Nov 29, 2022
Linear Algebra
tifrece2017
linear-algebra
eigen-values
+
–
1
votes
0
answers
343
TIFR ECE 2017 | Question: 8
Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$ \[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lfloor\frac{c}{n}\right\rfloor\] where $\lfloor t\rfloor$ denotes the ... $0$ The first sequence converges to $1 / c$, while the second sequence converges to $c$
Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$\[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lflo...
admin
46.4k
points
75
views
admin
asked
Nov 29, 2022
Others
tifrece2017
+
–
1
votes
0
answers
344
TIFR ECE 2017 | Question: 9
Recall that for a random variable $X$ which takes values in $\mathbb{N}$, the set of natural numbers, its entropy in bits is defined as \[H(X)=\sum_{n=1}^{\infty} p_{n} \log _{2} \frac{1}{p_{n}},\] where, for $n \in \mathbb{N}, p_{n}$ denotes the ... entropy of $X$ in bits? $1$ $1.5$ $\frac{1+\sqrt{5}}{2} \approx 1.618$ (the golden ratio) $2$ None of the above
Recall that for a random variable $X$ which takes values in $\mathbb{N}$, the set of natural numbers, its entropy in bits is defined as\[H(X)=\sum_{n=1}^{\infty} p_{n} \l...
admin
46.4k
points
73
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2017
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
345
TIFR ECE 2017 | Question: 10
Consider a single coin where the probability of heads is $p \in(0,1)$ and probability of tails is $1-p$. Suppose that this coin is flipped an infinite number of times. Let $N_{1}$ denote the number of flips till we see heads for the first time. Let $N_{2}$ denote the number of flips after ... $\frac{2}{p}$ $\frac{1}{p^{2}+(1-p)^{2}}$ $\frac{2}{p(1-p)}$
Consider a single coin where the probability of heads is $p \in(0,1)$ and probability of tails is $1-p$. Suppose that this coin is flipped an infinite number of times. Le...
admin
46.4k
points
91
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2017
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
346
TIFR ECE 2017 | Question: 11
Consider a unit length interval $[0,1]$ and choose a point $X$ on it with uniform distribution. Let $L_{1}=X$ and $L_{2}=1-X$ be the length of the two sub-intervals created by this point on the unit interval. Let $L=\max \left\{L_{1}, L_{2}\right\}$. Consider ... $\text{(ii)}$ Only $\text{(i)}$ and $\text{(iii)}$ Only $\text{(ii)}$ and $\text{(iv)}$ None of the above
Consider a unit length interval $[0,1]$ and choose a point $X$ on it with uniform distribution. Let $L_{1}=X$ and $L_{2}=1-X$ be the length of the two sub-intervals creat...
admin
46.4k
points
77
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2017
probability-and-statistics
probability
uniform-distribution
+
–
1
votes
0
answers
347
TIFR ECE 2017 | Question: 12
Consider a signal $X$ that can take two values, $-1$ with probability $p$ and $+1$ with probability $1-p$. Let $Y=X+N$, where $N$ is mean zero random noise that has probability density function symmetric about $0.$ Given $p$ and on observing $Y$, the detection problem is ... $\text{(iii)}$ Only $\text{(i)}$ and $\text{(ii)}$ Only $\text{(i)}$ and $\text{(iii)}$
Consider a signal $X$ that can take two values, $-1$ with probability $p$ and $+1$ with probability $1-p$. Let $Y=X+N$, where $N$ is mean zero random noise that has proba...
admin
46.4k
points
82
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2017
probability-and-statistics
probability
probability-density-function
+
–
1
votes
0
answers
348
TIFR ECE 2017 | Question: 13
Let $A$ be an $n \times n$ matrix. Consider the following statements. $A$ can have full-rank even if there exists two vectors $v_{1} \neq v_{2}$ such that $A v_{1}=A v_{2}$. $A$ can be similar to the identity matrix, when $A$ is not the identity matrix. Recall that ... $\text{(ii)}$ Only $\text{(iii)}$ $\text{(i), (ii),}$ and $\text{(iii)}$ None of the above
Let $A$ be an $n \times n$ matrix. Consider the following statements.$A$ can have full-rank even if there exists two vectors $v_{1} \neq v_{2}$ such that $A v_{1}=A v_{2}...
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46.4k
points
85
views
admin
asked
Nov 29, 2022
Linear Algebra
tifrece2017
linear-algebra
matrices
+
–
1
votes
0
answers
349
TIFR ECE 2017 | Question: 14
Consider the positive integer sequence \[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\] Which of the following statements is $\text{TRUE?}$ For every $M>0$, there exists an $n$ such that $x_{n}>M$ ... and then increases with $n \geq 1$ Sequence $\left\{x_{n}\right\}$ eventually converges to zero as $n \rightarrow \infty$ None of the above
Consider the positive integer sequence\[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\]Which of the following statements is $\text{TRUE?}$For every $M>0$, t...
admin
46.4k
points
88
views
admin
asked
Nov 29, 2022
Calculus
tifrece2017
calculus
maxima-minima
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–
1
votes
0
answers
350
TIFR ECE 2017 | Question: 15
Suppose that $f(x)$ is a real valued continuous function such that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$. Further, let \[a_{n}=\sum_{j=1}^{n} 1 / j\] and \[b_{n}=\sum_{j=1}^{n} 1 / j^{2} .\] Which of the following statements is true ... any number $M$ so that $f\left(b_{n}\right)$ and $f\left(a_{n}\right)$ are greater than $M$ for all $n$ None of the above
Suppose that $f(x)$ is a real valued continuous function such that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$. Further, let\[a_{n}=\sum_{j=1}^{n} 1 / j\]and\[b_{...
admin
46.4k
points
86
views
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asked
Nov 29, 2022
Others
tifrece2017
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0
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351
TIFR ECE 2016 | Question: 1
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible? $\int_{1}^{\infty} x f(x) \mathrm{d} x=\infty$ ... $\int_{1}^{\infty} \frac{f(x)}{1+\ln x} \mathrm{~d} x=\infty$ None of the above
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible?$\int_{1}^{\infty} ...
admin
46.4k
points
87
views
admin
asked
Nov 29, 2022
Calculus
tifrece2016
calculus
definite-integrals
+
–
1
votes
0
answers
352
TIFR ECE 2016 | Question: 2
Let $X_{1}$ and $X_{2}$ be two independent continuous real-valued random variables taking values in the unit interval $[0,1]$. Let $Y=\max \left\{X_{1}, X_{2}\right\}$ ... $\operatorname{Pr}[Z=1]>\operatorname{Pr}[Z=2]=\frac{1}{2}$ $\operatorname{Pr}[Z=1]<\operatorname{Pr}[Z=2]$
Let $X_{1}$ and $X_{2}$ be two independent continuous real-valued random variables taking values in the unit interval $[0,1]$. Let $Y=\max \left\{X_{1}, X_{2}\right\}$ an...
admin
46.4k
points
83
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2016
probability-and-statistics
probability
random-variable
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–
1
votes
0
answers
353
TIFR ECE 2016 | Question: 3
Let $(X, Y)$ be a pair of independent random variables. Suppose $X$ takes values in $\{1, \ldots, 6\}$ with equal probability, and $Y$ takes values in $\{2,3\}$ with $\operatorname{Pr}[Y=2]=p$. Let $Z=(X \bmod Y)+1$ ... $\operatorname{Pr}[Z=1]=\frac{1}{2}$ for $p=\frac{1}{2}$ $\operatorname{Pr}[Z=1]=p(1-p)$ None of the above
Let $(X, Y)$ be a pair of independent random variables. Suppose $X$ takes values in $\{1, \ldots, 6\}$ with equal probability, and $Y$ takes values in $\{2,3\}$ with $\op...
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354
TIFR ECE 2016 | Question: 4
Consider a system which in response to input $x(t)$ outputs \[y(t)=x\left(t^{2}\right) .\] Which of the following describes the system? linear, time-invariant, causal linear, time-invariant, non-causal linear, time-variant non-linear, time-invariant non-linear, time-variant
Consider a system which in response to input $x(t)$ outputs\[y(t)=x\left(t^{2}\right) .\]Which of the following describes the system?linear, time-invariant, causallinear,...
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355
TIFR ECE 2016 | Question: 5
Consider the opamp circuit in the figure. Approximately what is $V_{0}$? $-\left(\frac{V_{1}}{2}+V_{2}\right)$ $-\left(\frac{V_{1}}{4}+\frac{V_{2}}{2}\right)$ $-\left(V_{1}+2 V_{2}\right)$ $-\left(4 V_{1}+2 V_{2}\right)$ None of the above
Consider the opamp circuit in the figure. Approximately what is $V_{0}$?$-\left(\frac{V_{1}}{2}+V_{2}\right)$$-\left(\frac{V_{1}}{4}+\frac{V_{2}}{2}\right)$$-\left(V_{1}+...
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356
TIFR ECE 2016 | Question: 6
What is the Laplace transform $F(s)$ of the signal $f(t), t \geq 0$ defined below? In $t \in[0,1),$ ... $\frac{1}{s\left(1-e^{-s / 2}\right)}$ $\frac{1}{s\left(1+e^{-s / 2}\right)}$
What is the Laplace transform $F(s)$ of the signal $f(t), t \geq 0$ defined below? In $t \in[0,1),$\[f(t)=\left\{\begin{array}{ll}1, & t \in\left[0, \frac{1}{2}\right) \\...
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357
TIFR ECE 2016 | Question: 7
Suppose $X$ and $Y$ are independent Gaussian random variables, whose pdfs are represented below. Which of the following describes the pdf of the $X+Y?$
Suppose $X$ and $Y$ are independent Gaussian random variables, whose pdfs are represented below. Which of the following describes the pdf of the $X+Y?$
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TIFR ECE 2016 | Question: 8
In terms of their frequency responses, which of the following is the odd one out? All four circuits are equivalent
In terms of their frequency responses, which of the following is the odd one out? All four circuits are equivalent
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359
TIFR ECE 2016 | Question: 9
Suppose $Y=X+Z$, where $X$ and $Z$ are independent zero-mean random variables each with variance $1.$ Let $\hat{X}(Y)=a Y$ be the optimal linear least-squares estimate of $X$ from $Y$, i.e., $a$ is chosen such that $E\left[(X-a Y)^{2}\right]$ is minimized. What is the resulting ... $1$ $\frac{2}{3}$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$
Suppose $Y=X+Z$, where $X$ and $Z$ are independent zero-mean random variables each with variance $1.$ Let $\hat{X}(Y)=a Y$ be the optimal linear least-squares estimate of...
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TIFR ECE 2016 | Question: 10
Let $U_{1}, U_{2}, U_{3}$ be independent random variables that are each uniformly distributed between zero and one. What is the probability that the second highest value amongst the three lies between $1 / 3$ and $2 / 3?$ $\frac{2}{9}$ $\frac{1}{27}$ $\frac{13}{27}$ $\frac{1}{3}$ $\frac{7}{18}$
Let $U_{1}, U_{2}, U_{3}$ be independent random variables that are each uniformly distributed between zero and one. What is the probability that the second highest value ...
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