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Recent questions tagged expectation
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TIFR ECE 2014 | Question: 16
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is $4 \frac{1}{2}$ $3 \frac{1}{2}$ $5$ $4 \frac{17}{36} $ $4 \frac{3}{4}$
admin
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Probability and Statistics
Dec 14, 2022
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admin
44.9k
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10
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tifr2014
probability-and-statistics
probability
expectation
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2
TIFR ECE 2022 | Question: 1
Suppose that a random variable $X$ can take $5$ values $\{1,2,3,4,5\}$ with probabilities that depend upon $n \geq 0$ and are given by \[P(X=k)=\frac{e^{k n}}{e^{n}+e^{2 n}+e^{3 n}+e^{4 n}+e^{5 n}}\] for $k=1,2,3,4,5$. ... $1$ as $n \rightarrow \infty$ It converges to $5$ as $n \rightarrow \infty$ It converges to $0$ as $n \rightarrow \infty$
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Probability and Statistics
Nov 30, 2022
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admin
44.9k
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24
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tifrece2022
probability-and-statistics
probability
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3
TIFR ECE 2022 | Question: 11
A drunken man walks on a straight lane. At every integer time (in seconds) he moves a distance of $1$ unit randomly, either forwards or backwards. What is the expectation of the square of the distance after $100$ seconds from the initial position? Hint: ... sum of independent and identically distributed random variables. $100$ $\frac{\sqrt{300}}{4}$ $40$ $200$ $20 \pi$
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Probability and Statistics
Nov 30, 2022
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admin
44.9k
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21
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tifrece2022
probability-and-statistics
probability
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4
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 \[\mathbf{E}\left[\max \left(X_{1}, X_{2}\right) \mid \min \left(X_{1}, X_{2}\right)=3\right] ?\] $33 / 7$ $4$ $5$ $9 / 2$ $19 / 4$
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Probability and Statistics
Nov 30, 2022
by
admin
44.9k
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9
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tifrece2019
probability-and-statistics
probability
expectation
1
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5
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
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Nov 30, 2022
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8
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tifrece2019
probability-and-statistics
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1
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6
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? $7$ $5$ $1.5$ $3.5$ $2.5$
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Probability and Statistics
Nov 30, 2022
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admin
44.9k
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6
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tifrece2019
probability-and-statistics
probability
expectation
1
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7
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.) $\frac{a+b}{a+1}$ $\frac{a+b+1}{a}$ $\frac{a+b}{a}$ $\frac{a+b+1}{a+1}$ $a$
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Nov 30, 2022
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admin
44.9k
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8
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tifrece2019
probability-and-statistics
probability
expectation
1
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8
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
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Probability and Statistics
Nov 29, 2022
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44.9k
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25
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tifrece2018
probability-and-statistics
probability
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1
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9
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 \[E\left[X_{1} \mid \max \left(X_{1}, X_{2}\right)=5\right]\] $3$ $4$ $35 / 9$ $5 / 2$ $15 / 4$
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Nov 29, 2022
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admin
44.9k
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21
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tifrece2018
probability-and-statistics
probability
expectation
1
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10
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 ... $\frac{1}{1-(1-p)^{2}}(4+1 / p)$ $\frac{1}{p}+\frac{1}{1-p}$
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Nov 29, 2022
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44.9k
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tifrece2018
probability-and-statistics
probability
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11
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 ... $\frac{2}{p}$ $\frac{1}{p^{2}+(1-p)^{2}}$ $\frac{2}{p(1-p)}$
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Nov 29, 2022
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tifrece2017
probability-and-statistics
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12
TIFR ECE 2016 | Question: 11
Suppose that a random variable $X$ has a probability density function (pdf) given by \[f(x)=c \exp (-2 x)\] for $x \geq 1$, and $f(x)=0$, for $x<1$, where $c$ is an appropriate constant so that $f(x)$ is a valid pdf. What is the expected value of $X$ given that $X \geq 5?$ $5 \frac{1}{2}$ $7$ $10$ $8 \frac{1}{2}$ $6$
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probability-and-statistics
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GATE ECE 2020 | Question: 25
The two sides of a fair coin are labelled as $0$ to $1$. The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X = \text{min}(M, N)$, the expected value $E(X)$ (rounded off to two decimal places) is ___________.
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Probability and Statistics
Feb 13, 2020
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go_editor
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252
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gate2020-ec
numerical-answers
probability-and-statistics
probability
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0
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14
GATE ECE 2019 | Question: 18
If $X$ and $Y$ are random variables such that $E\left[2X+Y\right]=0$ and $E\left[X+2Y\right]=33$, then $E\left[X\right]+E\left[Y\right]=$___________.
Arjun
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Probability and Statistics
Feb 12, 2019
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Arjun
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96
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gate2019-ec
numerical-answers
probability-and-statistics
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0
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15
GATE ECE 2015 Set 3 | Question: 27
A fair die with faces $\{1, 2, 3, 4, 5, 6\}$ is thrown repeatedly till $’3’$ is observed for the first time. Let $X$ denote the number of times the die is thrown. The expected value of $X$ is _______.
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Mar 28, 2018
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gate2015-ec-3
numerical-answers
probability-and-statistics
probability
expectation
0
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16
GATE ECE 2015 Set 2 | Question: 29
Let the random variable $X$ represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of $X$ is _______.
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Mar 28, 2018
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gate2015-ec-2
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17
GATE ECE 2014 Set 3 | Question: 28
A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is _______ .
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Mar 26, 2018
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43
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gate2014-ec-3
probability-and-statistics
probability
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18
GATE ECE 2014 Set 1 | Question: 49
Let $X$ be a real-valued random variable with $E[X]$ and $E[X^{2}]$ denoting the mean values of $X$ and $X^{2},$ respectively. The relation which always holds true is $(E[X])^{2}>E[X^{2}]$ $E[X^{2}]\geq (E[X])^{2}$ $E[X^{2}] = (E[X])^{2}$ $E[X^{2}] > (E[X])^{2}$
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