Recent questions tagged tifrece2022

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...

Consider a coin flip game between Amar, Akbar and Anthony. A fair coin (so that heads and tails each have probability $0.5)$ is independently flipped five times. Amar win...

Consider two linear time invariant $\text{(LTI)}$ systems $T_{1}$ and $T_{2}$ with impulse responses $h_{1}(n)$ and $h_{2}(n)$, respectively. Let there be two cascades $C...

Evaluate the value of\[\max \left(x^{2}+(1-y)^{2}\right),\]where the maximisation above is over $x$ and $y$ such that $0 \leq x \leq y \leq 1$.$0$$2$$1 / 2$$1 / 4$$1$

Let $Q$ be a unit square in the plane with corners at $(0,0),(0,1),(1,0)$ and $(1,1)$. Let $B$ be a ball of radius $1$ in the plane centered at the origin $(0,0)$. Let $Q...

Consider a degree-$5$ polynomial function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$. If $f$ exhibits at least four local maxima, which of the following is necess...

Two players $\mathrm{A}$ and $\mathrm{B}$ of equal skill are playing a match. The first one to win $4$ rounds wins the match. Both players are equally likely to win each ...

Let $a, b, c$ be real numbers such that the following system of equations has a solution\[\begin{aligned}x+2 y+3 z &=a & & (1)\\8 x+10 y+12 z &=b & & (2)\\7 x+8 y+9 z &=c...

Suppose you throw a dart and it lands uniformly at random on a target which is a disk of unit radius. What is the probability density function $f(x)$ of the distance of t...

Find the vector which is closest (in Euclidean distance) to $\left(\begin{array}{lll}-1 & 1 & 1\end{array}\right)$ which can be written in the form\[a\left(\begin{array}{...

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...

An $n \times n$ matrix $\mathbf{P}$ is called a Permutation Matrix if each of its $n$ columns and $n$ rows contain exactly one $1$ and $n-1 \; 0$ 's. Consider the followi...

Calculate the minimum value attained by the function\[\sin (\pi x)-\sqrt{2} \pi x^{2}\]for values of $x$ which lie in the interval $[0,1]$.$\frac{1}{\sqrt{2}}\left(1-\fra...

Let a bag contain ten balls numbered $1,2, \ldots, 10$. Let three balls be drawn at random in sequence without replacement, and the number on the ball drawn on the $i^{\t...

Consider the difference below for $m \geq 5$:\[\sum_{n=1}^{m-1} \frac{1}{(1+n)^{2}}-\int_{x=1}^{m} \frac{1}{(1+x)^{2}} d x .\]Which statement about the difference is $\te...