## Marks 1

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

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 ...

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 (...

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}\,\,\,\,...

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

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

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

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 remove...

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...

$$X$$ 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?

## Marks 2

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 num...

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{...

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...

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 ga...

A random variable $$X$$ has probability density function $$f(x)$$ as given below:
$$$\,\,f\left( x \right) = \left\{ {\matrix{
{a + bx} & {for...

Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ two numbers are randomly selec...

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of these subjects, the student has ...

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 m...

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 w...

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

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

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...