Probability · Discrete Mathematics · GATE Data Science and Artificial Intelligence

Suppose $X$ and $Y$ are random variables. The conditional expectation of $X$ given $Y$ is denoted by $E[X \mid Y]$. Then $E[E[X \mid Y]]$ equals

Let X be a continuous random variable whose cumulative distribution function (CDF) $F_X(x)$, for some $t$, is given as follows:

$$ F_X(x)=\left\{\begin{array}{cc} 0 & x \leq t \\ \frac{x-t}{4-t} & t \leq x \leq 4 \\ 1 & x \geq 4 \end{array}\right. $$

If the median of X is 3 , then what is the value of $t$ ?

Let $X=a Z+b$, where Z is a standard normal random variable, and $a, b$ are two unknown constants. It is given that

$$ \begin{aligned} E[X] & =1, E[(X-E[X]) Z] \\ & =-2, E\left[(X-E[X])^2\right]=4 \end{aligned} $$

Where $E[X]$ denotes the expectation of random variable X . The values of $a, b$ are:

It is given that $P(X \geq 2)=0.25$ for an exponentially distributed random variable $X$ with $E[X]=\frac{1}{\lambda}$, where $E[X]$ denotes the expectation of $X$. What is the value of $\lambda$ ? (ln denotes natural logarithm)

There are three boxes containing white balls and black balls.

Box-1 contains 2 black and 1 white balls.

Box-2 contains 1 black and 2 white balls.

Box-3 contains 3 black and 3 white balls.

In a random experiment, one of these boxes is selected, where the probability of choosing Box-1 is $\frac{1}{2}$, Box-2 is $\frac{1}{6}$, and Box-3 is $\frac{1}{3}$. A ball is drawn at random from the selected box. Given that the ball drawn is white, the probability that it is drawn from Box-2 is ____________. (Round off to two decimal places)

Let $Y=Z^2, Z=\frac{X-\mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is

Consider the cumulative distribution function (CDF) of a random variable X :

$$ F_X(x)=\left\{\begin{array}{cc} 0 & x \leq-1 \\ \frac{1}{4}(x+1)^2 & -1 \leq x \leq 1 \\ 1 & x \geq 1 \end{array}\right. $$

The value of $P\left(X^2 \leq 0.25\right)$

A random variable X is said to be distributed as $\operatorname{Bernoulli}(\theta)$, denoted by $X \sim \operatorname{Bernoulli}(\theta)$, if

$$ P(X=1)=\theta, P(X=0)=1-\theta $$

for $0<\theta<1$. Let $Y=\sum_{i=1}^{300} X_i$. Where $X_i \sim \operatorname{Bernoulli}(\theta), i=1,2, \ldots \ldots, 300$ be independent and identically distributed random variables with $\theta=0.25$. The value of $P(60 \leq \mathrm{Y} \leq 90)$, after approximation through Central Limit Theorem, is given by

(Recall that $\phi(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{t^2}{2}} d t$ )

For $x \in \mathbb{R}$, the floor function is denoted by $f(x)=\lfloor x\rfloor$ and defined as follows $\lfloor x\rfloor=k, k \leq x

where $k$ is an integer. Let $Y=\lfloor X\rfloor$, where $X$ is an exponentially distributed random variable with mean $\frac{1}{\ln 10}$, where In denotes natural logarithm. For any positive integer $l$, one can write the probability of the event $Y=l$ as follows

$$ P(Y=l)=q^l(1-q) $$

The value of $q$ is

Consider a coin-toss experiment where the probability of head showing up is $p$. In the $i^{\text {th }}$ coin toss, let $X_i=1$ if head appears, and $X_i=0$ if tail appears.

Consider

$$ \hat{p}=\frac{1}{n} \sum_{i=1}^n X_i $$

where $n$ is the total number of independent coin tosses.

A bag contains 5 white balls and 10 black balls. In a random experiment, $n$ balls are drawn from the bag one at a time with replacement. Let $S_n$ denote the total number of black balls drawn in the experiment.

The expectation of $S_{100}$ denoted by $E\left[S_{100}\right]=$ ___________ (Round off to one decimal place)