Probability & Stats (EE)

Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y=X^2$ and $P_X(x)=\left\{\begin{array}{cc}1, & x \in(0,1] \\ 0, & \text { otherwise }\...

Scores obtained by two students P and Q in seven courses are given in the table below. Based on the information given in the table, which one of the following statements is INCORRECT? $$ \begin{array}{|l|l|l|l|l|l|l|l|}...

Consider discrete random variable $X$ and $Y$ with probabilities as follows: $$ \begin{aligned} & P(X=0 \text { and } Y=0)=\frac{1}{4} \\ & P(X=1 \text { and } Y=1)=\frac{1}{8} \\ & P(X=0 \text { and } Y=1)=\frac{1}{2} \...

Let $X$ be a discrete random variable that is uniformly distributed over the set {$-10, -9, \cdots, 0, \cdots, 9, 10$}. Which of the following random variables is/are uniformly distributed?

One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation $$\sigma_0$$. The $$\sigma_0$$ is estimated by randomly drawing out 10,0...

Given a fair six-faced dice where the faces are labelled '1', '2', '3', '4', '5', and '6', what is the probability of getting a '1' on the first roll of the dice and a '4' on the second roll?

The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is __________ (Round off to 3 decimal places).

Let the probability density function of a random variable x be given as f(x) = ae $$-$$2|x| The value of a is _________.

There are two identical dice with a single letter on each of the faces. The following six letters : Q, R, S, T, U and V, one on each of the faces. Any of the six outcomes are equally likely. The two dice are thrown once...

Suppose the probability that a coin toss shows "head" is $p$, where $0

Let $X$ is a continuous random variable denoting the temperature measured. The range of temperature is $[0,100]$ degree Celsius and let the probability density function of $X$ be $f(x)=0.01$ for $0 \leq X \leq 100$. The...

How many integers are there between 100 and 1000 all of whose digits are even?

A class of twelve children has two more boys than girls. A group of three children are randomly picked from this class to accompany the teacher on a field trip. What is the probability that the group accompanying the tea...

An e-mail password must contain three characters. The password has to contain one numeral from 0 to 9, one upper case and one lower case character from the English alphabet. How many distinct passwords are possible?

A person decides to toss a fair coin repeatedly until he gets a head. He will make at most $$3$$ tosses. Let the random variable $$Y$$ denotes the number of heads. The value of var $$\left\{ Y \right\},$$ where var $$\le...

An urn contains $$5$$ red balls and $$5$$ black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball in the second draw is

Assume that in a traffic junction, the cycle of the traffic signal lights is $$2$$ minutes of green (vehicle does not stop) and $$3$$ minutes of red (vehicle stops). Consider that the arrival time of vehicles at the junc...

Let the probability density function of a random variable $$X,$$ be given as: $$${f_x}\left( x \right) = {3 \over 2}{e^{ - 3x}}u\left( x \right) + a{e^{4x}}u\left( { - x} \right)$$$ where $$u(x)$$ is the unit step functi...

Candidates were asked to come to an interview with $$3$$ pens each. Black, blue, green and red were the permitted pen colours that the candidate could bring. The probability that a candidate comes with all $$3$$ pens hav...

Two players, $$A$$ and $$B,$$ alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player $$A$$ starts the game, the probability that $$A$$ wins the game is

Two coins $$R$$ and $$S$$ are tossed. The $$4$$ joint events $$\,\,\,\,\,\,{H_R}{H_S},\,\,\,\,{T_R}{T_S},\,\,\,\,{H_R}{T_S},\,\,\,\,{T_R}{H_S}\,\,\,\,\,\,\,$$ have probabilities $$0.28,$$ $$0.18,$$ $$0.30,$$ $$0.24$$ res...

Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ two numbers are randomly selected, one from each set. What is the probability that the sum of the tw...

A random variable $$X$$ has probability density function $$f(x)$$ as given below: $$$\,\,f\left( x \right) = \left\{ {\matrix{ {a + bx} & {for\,\,0 < x < 1} \cr 0 & {otherwise} \cr } } \right.\,\,$$$ If the expected valu...

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of these subjects, the student has $$75$$% chance of passing in at least one, a $$50$$% chance of passing...

Let $$X$$ be a random variable with probability density function $$f\left( x \right) = \left\{ {\matrix{ {0.2} & {for\,\left| x \right| \le 1} \cr {0.1} & {for\,1 < \left| x \right| \le 4} \cr 0 & {otherwise} \cr } } \ri...

A fair coin is tossed $$n$$ times. The probability that the difference between the number of heads and tails is $$(n-3)$$ is

Consider a die with the property that the probability of a face with $$'n'$$ dots showing up is proportional to $$'n'.$$ The probability of the face with three dots showing up is _________.

Lifetime of an electric bulb is a random variable with density $$f\left( x \right) = k{x^2},$$ where $$x$$ is measured in years. If the minimum and maximum lifetimes of bulb are $$1$$ and $$2$$ years respectively, then t...

A continuous random variable $$X$$ has a probability density function $$f\left( x \right) = {e^{ - x}},0 < x < \infty .$$ Then $$P\left\{ {X > 1} \right\}$$ is

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

Two independent random variables $$X$$ and $$Y$$ are uniformly distributed in the interval $$\left[ { - 1,1} \right].$$ The probability that max $$\left[ {X,Y} \right]$$ is less than $$1/2$$ is

A box contains $$4$$ white balls and $$3$$ red balls. In succession, two balls are randomly selected and removed from the box. Given that first removed ball is white, the probability that the $$2$$ nd removed ball is red...

Assume for simplicity that $$N$$ people, all born in April (a month of $$30$$ days) are collected in a room, consider the event of at least two people in the room being born on the same date of the month (even if in diff...

$$X$$ is uniformly distributed random variable that take values between $$0$$ and $$1.$$ The value of $$E\left( {{X^3}} \right)$$ will be

If $$P$$ and $$Q$$ are two random events, then which of the following is true?

A fair coin is tossed $$3$$ times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is