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TIFR ECE 2023 | Question: 1
Consider a fair coin with probability of heads and tails equal to $1 / 2$. Moreover consider two dice, first $\mathrm{D}_{1}$ that has three faces numbered $1,3,5$ and second $\mathrm{D}_{2}$ that has three faces numbered $2,4,6$ ... rolled dice in the experiment. What is $\mathbb{E}[X]$ ? $\frac{7}{2}$ 4 3 $\frac{9}{2}$ None of the above

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TIFR ECE 2023 | Question: 2
$\begin{array}{rlr}a^*=\max _{x, y} & x^2+y^2-8 x+7 \\ \text { s.t. } & x^2+y^2 \leq 1 \\ & y \geq 0\end{array}$ Then $a^{\star}$ is $16$ $14$ $12$ $10$ None of the above

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TIFR ECE 2023 | Question: 3
Let \[ \mathcal{P}=\left\{(x, y): x+y \geq 1,2 x+y \geq 2, x+2 y \geq 2,(x-1)^{2}+(y-1)^{2} \leq 1\right\} . \] Compute \[ \min _{(x, y) \in \mathcal{P}} 2 x+3 y \] $2$ $3$ $4$ $6$ None of the above

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TIFR ECE 2023 | Question: 4
Recall that the entropy (in bits) of a random variable $\mathrm{X}$ which takes values in $\mathbb{N}$ ... random variable which denotes the number of tosses made. What is the entropy of $\mathrm{X}$ in bits? $1$ $2$ $4$ Infinity None of the above

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5

TIFR ECE 2023 | Question: 5
Let $\mathrm{B}$ denote the unit ball in $\mathbb{R}^{2}$, and $\mathrm{Q}$ a square of side length $2$. Let $\mathrm{K}$ be the set of all vectors $z$ such that for some $x \in \mathrm{B}$ and some $y \in \mathrm{Q}, z=x+y$. The area of $\mathrm{K}$ is $4+\pi$ $6+\pi$ $8+\pi$ $10+\pi$ $12+\pi$

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6

TIFR ECE 2023 | Question: 6
An ant in the plane travels in a spiral such that its position $(x(t), y(t))$ at time $t \geq 0$ is $\left(e^{t} \cos t, e^{t} \sin t\right)$. At time $t=1$, find the real part of $\ln (x(t)+i y(t))$. $-2$ $1$ $0$ $-1$ $2$

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TIFR ECE 2023 | Question: 7
Let $f(x)$ be a positive continuous function on the real line that is the density of a random variable $X$. The differential entropy of $X$ is defined to be $-\int_{-\infty}^{\infty} f(x) \ln f(x) d x$. In which case does $X$ have the least differential entropy? You may use these facts: The ... $f(x):=(1 / 4) e^{-|x| / 2}$. $f(x):=e^{-2|x|}$.

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TIFR ECE 2023 | Question: 8
Suppose a bag contains 5 red balls, 3 blue balls, and 2 black balls. Balls are drawn without replacement until the bag is empty. Let $X_{i}$ be a random variable which takes value 1 if the $i$-th ball drawn is red, value 2 if that ball is blue, and 3 if it is black. Let the ... Only (i) and (ii) Only (i) and (iii) All of (i), (ii), and (iii) None of (i), (ii), or (iii)

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TIFR ECE 2023 | Question: 9
Consider an $n \times n$ matrix $A$ with the property that each element of $A$ is non-negative and the sum of elements of each row is $1$ . Consider the following statements. 1. $1$ is an eigenvalue of $A$ 2. The magnitude of any eigenvalue of $A$ ... statements $1$ and $3$ are correct Only statements $2$ and $3$ are correct All statements $1,2$ , and $3$ are correct

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10

TIFR ECE 2023 | Question: 10
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows: \[ f(t) * g(t)=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau \] Let $u(t)$ be the unit-step function, i.e., $u(t)=1$ for $t \geq 0$ and $u(t)=0$ for $t<0$. What is $f(t) * g(t)$ ... $\frac{1}{2}(\exp (-t)+\sin (t)-2 \cos (t)) u(t)$ $\frac{1}{2}(\exp (-t)-\sin (t)+2 \cos (t)) u(t)$

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11

TIFR ECE 2023 | Question: 11
Consider the function \[ f(x)=x e^{|x|}+4 x^{2} \] for values of $x$ which lie in the interval $[-1,1]$. In this domain, suppose the function attains the minimum value at $x^{*}$. Which of the following is true? $-1 \leq x^{*}<-0.5$ $-0.5 \leq x^{*}<0$ $x^{*}=0$ $0<x^* \leq 0.5$ $0.5<x^* \leq 1$

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12

TIFR ECE 2023 | Question: 12
Consider a disk $D$ of radius 1 centered at the origin. Let $X$ be a point uniformly distributed on $D$ and let the distance of $X$ from the origin be $R$. Let $A$ be the (random) area of the disk with radius $R$ centered at the origin. Then $\mathbb{E}[A]$ is $\frac{\pi}{3}$ $\frac{\pi}{6}$ $\frac{\pi}{4}$ $\frac{\pi}{2}$ None of the above

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13

TIFR ECE 2023 | Question: 13
Let $X$ be a random variable which takes values $1$ and $-1$ with probability $1 / 2$ each. Suppose $Y=X+N$, where $N$ is a random variable independent of $X$ with the following probability density function (p.d.f.): \[ f_{N}(n)=\left\{\begin{array} ... $0$ $1 / 8$ $1 / 4$ $1 / 2$ None of the above

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14

TIFR ECE 2023 | Question: 14
Suppose that $Z \sim \mathcal{N}(0,1)$ is a Gaussian random variable with mean zero and variance $1$. Let $F(z) \equiv \mathbb{P}(Z \leq z)$ be the cumulative distribution function $\operatorname{(CDF)}$ of $Z$. Define a new random variable $Y$ as $Y=F(Z)$. This means that the ... of $\mathbb{E}[Y]$ is: $F(1)$ $1$ $\frac{1}{2}$ $\frac{1}{\sqrt{2 \pi}}$ $\frac{\pi}{4}$

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15

TIFR ECE 2023 | Question: 15
Let $\left\{x_{n}\right\}_{n \geq 0}$ be a sequence of real numbers which satisfy \[ x_{n+1}\left(1+x_{n+1}\right) \leq x_{n}\left(1+x_{n}\right), \quad n \geq 0 . \] Choose the correct option from the ... is always bounded but does not necessarily converge. The sequence always converges to a non-zero limit. The sequence always converges to zero. None of the above.

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