Recent questions tagged tifrece2020

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...

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 ...

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...

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...

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...

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...

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...

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...

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...

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...

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) /...

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...

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...

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...

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 $\...