Questions by admin / GO Electronics

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281

TIFR ECE 2020 | 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, \\ \alpha y[n-1] ... Non-linear, time-invariant, BIBO stable Linear, time-variant, BIBO unstable Non-linear, time-variant, BIBO stable 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...

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asked Nov 30, 2022

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282

TIFR ECE 2020 | Question: 2
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows: $f(t) * g(t)=$ $\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$. If $f(t) * g(t)=h(t)$, what is $f(t-1) * g(t+1)?$ $h(2 t)$ $h(t)$ $h(t-1)$ $h(t+1)$ None of the above

Convolution between two functions $f(t)$ and $g(t)$ is defined as follows: $f(t) * g(t)=$ $\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$. If $f(t) * g(t)=h(t)$, what ...

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asked Nov 30, 2022

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283

TIFR ECE 2020 | Question: 3
Balls are drawn one after the other uniformly at random without replacement from a set of eight balls numbered $1,2, \ldots, 8$ until all balls drawn. What is the expected number of balls whose value match their ordinality (i.e., their position in the order in which ... ? Now can you use linearity of expectation to solve the problem? $1$ $1.5$ $2$ $2.5$ None of the above

Balls are drawn one after the other uniformly at random without replacement from a set of eight balls numbered $1,2, \ldots, 8$ until all balls drawn. What is the expecte...

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asked Nov 30, 2022

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284

TIFR ECE 2020 | Question: 4
Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions that are continuous and differentiable. Consider the following statements: $\min \{f, g\}$ is continuous $\max \{f, g\}$ is continuous $\max \{f, g\}$ is differentiable Which ... is correct Only statement $2$ is correct Only statement $3$ is correct Only statements $1$ and $2$ are correct None of the above

Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions that are continuous and differentiable. Consider the following statements:$\min \{f, g\}$ is continuous$\ma...

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asked Nov 30, 2022

1 votes

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285

TIFR ECE 2020 | Question: 5
Let $f(t)$ be a periodic signal of period $1$, i.e. $f(t+1)=f(t) \forall t$. Define the averaging operator depending on a fixed parameter $h>0$ as below: \[g(x)=\frac{1}{2 h} \int_{x-h}^{x+h} f(t) d t .\] Which of the following is ... $\frac{1}{2}$ $g(x)$ is periodic with period $1$ The value of $h$ determines whether or not $g(x)$ is periodic None of the above

Let $f(t)$ be a periodic signal of period $1$, i.e. $f(t+1)=f(t) \forall t$. Define the averaging operator depending on a fixed parameter $h>0$ as below:\[g(x)=\frac{1}{2...

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asked Nov 30, 2022

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286

TIFR ECE 2020 | Question: 6
For all values of $r>0$, the area of the set of all points outside the unit square whose Euclidean distance to the unit square is less than $r$ is: $=\pi r^{2}+4 r$ $<4 \pi r^{2}$ $>4 \pi r^{3}+4 r$ $=\frac{4 \pi r^{3}}{3}+6 r+2 \pi r^{2}$ None of the above

For all values of $r>0$, the area of the set of all points outside the unit square whose Euclidean distance to the unit square is less than $r$ is:$=\pi r^{2}+4 r$$<4 \pi...

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asked Nov 30, 2022

1 votes

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287

TIFR ECE 2020 | Question: 7
Given $n$ independent Bernoulli random variables, taking value $1$ with probability $p$ and $0$ with probability $1-p$. Then, which of the following is the value of $E\left[\left(z_{1}+\right.\right.$ $\left.\left.\ldots z_{n}\right)^{2}\right] ?$ $0$ $n p+n(n-1) p^{2}$ $n^{3} p^{2}$ $n^{2} p^{2}+n p$ None of the above

Given $n$ independent Bernoulli random variables, taking value $1$ with probability $p$ and $0$ with probability $1-p$. Then, which of the following is the value of $E\le...

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asked Nov 30, 2022

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288

TIFR ECE 2020 | Question: 8
Suppose that Dice $1$ has five faces numbered $1$ to $5,$ each of which is equally likely to occur once the dice is rolled. Dice $2$ similarly has eight equally likely faces numbered $1$ to $8.$ Suppose that the two dice are rolled, and the sum is equal to $8.$ Conditioned on this, ... $2?$ $1 / 4$ $1 / 3$ $1 / 2$ $2 / 7$ $2 / 5$

Suppose that Dice $1$ has five faces numbered $1$ to $5,$ each of which is equally likely to occur once the dice is rolled. Dice $2$ similarly has eight equally likely fa...

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asked Nov 30, 2022

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289

TIFR ECE 2020 | Question: 9
Let $A$ be an $n \times n$ matrix with the the property that $A^{m}=0$ for some $m \in \mathbb{N}$. Consider the following statements: At least one entry of $A$ is zero All eigenvalues of $A$ are zero All diagonal entries of $A$ are zero ... $2$ alone is correct Only statement $3$ is correct Only statements $1$ and $2$ are correct Only statements $2$ and $3$ are correct

Let $A$ be an $n \times n$ matrix with the the property that $A^{m}=0$ for some $m \in \mathbb{N}$. Consider the following statements:At least one entry of $A$ is zeroAll...

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asked Nov 30, 2022

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290

TIFR ECE 2020 | Question: 10
Consider two independent random variables $\left(U_{1}, U_{2}\right)$ both are uniformly distributed between $[0,1]$. The conditional expectation \[E\left[\left(U_{1}+U_{2}\right) \mid \max \left(U_{1}, U_{2}\right) \geq 0.5\right]\] equals $7 / 6$ $8 / 7$ $6 / 7$ $1.1$ None of the above

Consider two independent random variables $\left(U_{1}, U_{2}\right)$ both are uniformly distributed between $[0,1]$. The conditional expectation\[E\left[\left(U_{1}+U_{2...

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asked Nov 30, 2022

1 votes

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291

TIFR ECE 2020 | Question: 11
Suppose that $X$ is a real valued random variable and $E[\exp X]=2$. Then, which of the following must be $\text{TRUE? Hint:}$ $(\exp (x)+\exp (y)) / 2 \geq \exp ((x+y) / 2)$. $E[X]<\ln 2$ $E[X]>\ln 2$ $E[X] \geq \ln 2$ $E[X] \leq \ln 2$ None of the above

Suppose that $X$ is a real valued random variable and $E[\exp X]=2$. Then, which of the following must be $\text{TRUE? Hint:}$ $(\exp (x)+\exp (y)) / 2 \geq \exp ((x+y) /...

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asked Nov 30, 2022

1 votes

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292

TIFR ECE 2020 | Question: 12
Consider a unit disc $D$. Let a point $x$ be chosen uniformly on $D$ and let the random distance to $x$ from the center of $D$ be $R$. Which of the following is $\text{TRUE?}$ $R^{2}$ is uniformly distributed in $[0,1]$ $\pi R^{2}$ is uniformly ... $[0,1]$ $2 \pi R^{2}$ is uniformly distributed in $[0,1]$ None of the above

Consider a unit disc $D$. Let a point $x$ be chosen uniformly on $D$ and let the random distance to $x$ from the center of $D$ be $R$. Which of the following is $\text{TR...

33 views

asked Nov 30, 2022

1 votes

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293

TIFR ECE 2020 | Question: 13
Alice and Bob have one coin each with probability of Heads $p$ and $q$, respectively. In each round, both Alice and Bob independently toss their coin once, and the game stops if one of them gets a Heads and the other gets a Tails. If they both get either Heads or both get Tails in ... $R$ is independent of $p$ and $q$ $R=\frac{1}{1+2 p q-p-q}$ None of the above

Alice and Bob have one coin each with probability of Heads $p$ and $q$, respectively. In each round, both Alice and Bob independently toss their coin once, and the game s...

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asked Nov 30, 2022

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294

TIFR ECE 2020 | Question: 14
Two matrices $A$ and $B$ are called similar if there exists an invertible matrix $X$ such that $A=X^{-1} B X$. Let $A$ and $B$ be two similar matrices. Consider the following statements: $\operatorname{det}(x I-A)=\operatorname{det}(x I-B)$ ... statement $2$ is correct Only statements $1$ and $2$ are correct All Statements $1, 2$ and $3$ are correct None of the above

Two matrices $A$ and $B$ are called similar if there exists an invertible matrix $X$ such that $A=X^{-1} B X$. Let $A$ and $B$ be two similar matrices. Consider the follo...

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asked Nov 30, 2022

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295

TIFR ECE 2020 | Question: 15
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors such that $\vec{v}_{1}^{T} \vec{v}_{2}=0$. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such that they solve the following optimization problem \[ ... $\left\|\vec{v}_{*}\right\|^{2}-\|\vec{u}\|^{2}$ $0$ None of the above

Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors such that $\vec{v}_{1}^{T} \vec{v}_{2}=0$. Let the pair of real numbers $\...

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asked Nov 30, 2022

1 votes

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296

TIFR ECE 2019 | 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, \\ \alpha y[2 n-1]+\beta ... -linear, but time-invariant only if $\alpha=0$ (parameters $\beta$ and $\gamma$ can take arbitrary values) 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...

39 views

asked Nov 30, 2022

1 votes

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297

TIFR ECE 2019 | Question: 2
Let $A$ and $B$ be two square matrices that have full rank. Let $\lambda_{A}$ be an eignevalue of $A$ and $\lambda_{B}$ an eigenvalue of $B$. Which of the following is always $\text{TRUE}?$ $A B$ has full rank $A-B$ ... an eigenvalue of $A B$ $A+B$ has full rank At least one of $\lambda_{A}$ or $\lambda_{B}$ is an eigenvalue of $A B$

Let $A$ and $B$ be two square matrices that have full rank. Let $\lambda_{A}$ be an eignevalue of $A$ and $\lambda_{B}$ an eigenvalue of $B$. Which of the following is al...

29 views

asked Nov 30, 2022

1 votes

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298

TIFR ECE 2019 | Question: 3
Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which of the following is $\text{TRUE}?$ $\lim _{x \rightarrow \infty} f(x)=1$ ... $1-\epsilon$ $\max _{x \geq 1} f(x)=1$ None of the above

Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which...

33 views

asked Nov 30, 2022

1 votes

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299

TIFR ECE 2019 | Question: 4
Let $f(x)=\sqrt{x^{2}-4 x+4},$ for $x \in(-\infty, \infty)$. Here, $\sqrt{y}$ denotes the non-negative square root of $y$ when $y$ is non-negative. Then, which of the following is $\text{TRUE}?$ $f(x)$ is ... differentiable $f(x)$ is continuous and differentiable $f(x)$ is continuous but not differentiable $f(x)$ is neither continuous nor differentiable None of the above

Let $f(x)=\sqrt{x^{2}-4 x+4},$ for $x \in(-\infty, \infty)$. Here, $\sqrt{y}$ denotes the non-negative square root of $y$ when $y$ is non-negative. Then, which of the fol...

29 views

asked Nov 30, 2022

1 votes

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300

TIFR ECE 2019 | Question: 5
Consider the function $f(x)=e^{x^{2}}-8 x^{2}$ for all $x$ on the real line. For how many distinct values of $x$ do we have $f(x)=0?$ $1$ $4$ $2$ $3$ $5$

Consider the function $f(x)=e^{x^{2}}-8 x^{2}$ for all $x$ on the real line. For how many distinct values of $x$ do we have $f(x)=0?$ $1$$4$$2$$3$$5$

31 views

asked Nov 30, 2022

1 votes

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301

TIFR ECE 2019 | Question: 6
Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice. Each of the dice takes each of the six values $1,2,3,4,5$, and $6$ with equal probability. What is the value of the conditional expectation \[\mathbf{E}\left[\max \left(X_{1}, X_{2}\right) \mid \min \left(X_{1}, X_{2}\right)=3\right] ?\] $33 / 7$ $4$ $5$ $9 / 2$ $19 / 4$

Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice. Each of the dice takes each of the six values $1,2,3,4,5$, and $6$ with equa...

32 views

asked Nov 30, 2022

1 votes

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302

TIFR ECE 2019 | Question: 7
Consider two random variables $X$ and $Y$ which take values in a finite set $S$. Let $p_{X, Y}$ represent their joint probability mass function (p.m.f.) and let $p_{X}$ and $p_{Y}$, respectively, be the marginal p.m.f.'s of $X$ and $Y$, respectively. Which of ... None of the above

Consider two random variables $X$ and $Y$ which take values in a finite set $S$. Let $p_{X, Y}$ represent their joint probability mass function (p.m.f.) and let $p_{X}$ a...

29 views

asked Nov 30, 2022

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303

TIFR ECE 2019 | Question: 8
Let $K$ be a cube of side $1$ in $\mathbb{R}^{3}$, with its centre at the origin, and its sides parallel to the co-ordinate axes. For $t \geq 0$, let $K_{t}$ be the set of all points in $\mathbb{R}^{3}$ whose Euclidean distance to $K$ is less than or equal to $t$ ... $V \leq \frac{4}{3} \pi\left(\frac{\sqrt{3}}{2}+t\right)^{3}$ $V \geq(1+2 t)^{3}$

Let $K$ be a cube of side $1$ in $\mathbb{R}^{3}$, with its centre at the origin, and its sides parallel to the co-ordinate axes. For $t \geq 0$, let $K_{t}$ be the set o...

32 views

asked Nov 30, 2022

1 votes

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304

TIFR ECE 2019 | Question: 9
Consider a coin which comes up heads with probability $p$ and tails with probability $1-p$, where $0 < p < 1.$ Suppose we keep tossing the coin until we have seen both sides of the coin. What is the expected number of times we would have seen tails? (Hint: the expected number of ... $(1/p.)$ $\frac{1}{p}$ $1+\frac{1}{1-p}$ $p+\frac{1}{p}-1$ $2$ None of the above

Consider a coin which comes up heads with probability $p$ and tails with probability $1-p$, where $0 < p < 1.$ Suppose we keep tossing the coin until we have seen both si...

34 views

asked Nov 30, 2022

1 votes

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305

TIFR ECE 2019 | Question: 10
Let $X, Z_{1}$, and $Z_{2}$ be independent random variables taking values in the set $\{0,1\}$. $X$ is uniformly distributed in $\{0,1\}$, while the distributions of $Z_{1}$ and $Z_{2}$ are such that if we define $Y_{1}=X+Z_{1}$ and $Y_{2}=X+Z_{2}$, where addition ... $\left(1 / p_{1}+1 / p_{2}\right)^{-1}$ $\left(1+1 / p_{1}+1 / p_{2}\right)^{-1}$ None of the above

Let $X, Z_{1}$, and $Z_{2}$ be independent random variables taking values in the set $\{0,1\}$. $X$ is uniformly distributed in $\{0,1\}$, while the distributions of $Z_{...

32 views

asked Nov 30, 2022

1 votes

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306

TIFR ECE 2019 | Question: 11
Let $X$ and $Y$ be independent Gaussian random variables with means $1$ and $2$ and variances $3$ and $4$ respectively. What is the minimum possible value of $\mathbf{E}\left[(X+Y-t)^{2}\right]$, when $t$ varies over all real numbers? $7$ $5$ $1.5$ $3.5$ $2.5$

Let $X$ and $Y$ be independent Gaussian random variables with means $1$ and $2$ and variances $3$ and $4$ respectively. What is the minimum possible value of $\mathbf{E}\...

34 views

asked Nov 30, 2022

1 votes

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307

TIFR ECE 2019 | Question: 12
Consider an urn with $a$ red and $b$ blue balls. Balls are drawn out one-by-one, without replacement and uniformly at random, until the first red ball is drawn. What is the expected total number of balls drawn by this process? (Hint: Consider deriving an appropriate recurrence.) $\frac{a+b}{a+1}$ $\frac{a+b+1}{a}$ $\frac{a+b}{a}$ $\frac{a+b+1}{a+1}$ $a$

Consider an urn with $a$ red and $b$ blue balls. Balls are drawn out one-by-one, without replacement and uniformly at random, until the first red ball is drawn. What is t...

32 views

asked Nov 30, 2022

1 votes

0 answers

308

TIFR ECE 2019 | Question: 13
For $t>0$, let $S_{t}$ denote the ball of radius $t$ centered at the origin in $\mathbb{R}^{n}$. That is, \[S_{t}=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \sum_{i=1}^{n} x_{i}^{2} \leq t^{2}\right\} .\] Let $N_{t}$ be the number of ... $\lim _{t \rightarrow \infty} \frac{N_{t}}{V_{t}}=1$ $N_{t}$ is a monotonically decreasing function of $t$

For $t>0$, let $S_{t}$ denote the ball of radius $t$ centered at the origin in $\mathbb{R}^{n}$. That is,\[S_{t}=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \sum_{i=1}^{n} ...

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asked Nov 30, 2022

1 votes

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309

TIFR ECE 2019 | Question: 14
Consider the circle of radius $1$ centred at the origin in two dimensions. Choose two points $x$ and $y$ independently at random so that both are uniformly distributed on the circle. Let the vectors joining the origin to $x$ and $y$ be $X$ and $Y$, respectively. Let $\theta$ be ... $\mathbf{E}\left[|x-y|^{2}\right]=\sqrt{3}$ $\mathbf{E}\left[|x-y|^{2}\right]=1$

Consider the circle of radius $1$ centred at the origin in two dimensions. Choose two points $x$ and $y$ independently at random so that both are uniformly distributed on...

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asked Nov 30, 2022

1 votes

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310

TIFR ECE 2019 | Question: 15
Anu reached a bus stop at $\text{9:00 AM.}$ She knows that the number of minutes after $\text{9:00 AM}$ when the bus will arrive is distributed with probability density function (p.d.f.) $f$ where \[f(x)=\frac{1}{10} \exp (-x / 10)\] for $x \geq 0$, ... time, measured in minutes after $\text{9:00 AM,}$ would Anu expect the bus to arrive? $12.5$ $15$ $7.5$ $10$ $12.5$

Anu reached a bus stop at $\text{9:00 AM.}$ She knows that the number of minutes after $\text{9:00 AM}$ when the bus will arrive is distributed with probability density f...

34 views

asked Nov 30, 2022

1 votes

0 answers

311

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...

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asked Nov 29, 2022

1 votes

0 answers

312

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...

90 views

asked Nov 29, 2022

1 votes

0 answers

313

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...

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asked Nov 29, 2022

1 votes

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314

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 $f(x)$ ...

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asked Nov 29, 2022

1 votes

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315

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 ...

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asked Nov 29, 2022

1 votes

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316

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...

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asked Nov 29, 2022

1 votes

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317

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}, ...

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asked Nov 29, 2022

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318

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...

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asked Nov 29, 2022

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319

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 $Z_{1}=\max (X, Y)$...

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asked Nov 29, 2022

1 votes

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320

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 ...

100 views

asked Nov 29, 2022