GATE CS Probability: Random Variables and Distributions questions

Practise Probability: Random Variables and Distributions questions for GATE CS. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

Define the sample space and events first. Distinguish independent events from conditional probabilities, and check that the final probability lies between zero and one.

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SubjectTopicYearType

An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is put back in the urn. The probability that there are equal number of red and blue balls after two steps is

X

be a random variable which takes values in the set

\{1,2,3,4,5,6,7,8\}

\operatorname{Pr}(X=1)=\operatorname{Pr}(X=2)=\operatorname{Pr}(X=5)=\operatorname{Pr}(X=7)=\frac{1}{6}

\operatorname{Pr}(X=3)=\operatorname{Pr}(X=4)=\operatorname{Pr}(X=6)=\operatorname{Pr}(X=8)=\frac{1}{12}

. The expected value of

X

, denoted by …

The probability density function

f(x)

of a random variable

X

which takes real values is

f(x)=\frac{1}{3 \sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{18}\right), \quad x \in(-\infty,+\infty)

Which one of the following statements is correct about the random variable

X

5

4

regular coins and

1

fake coin. When a regular coin is tossed, the probability

P(

)=0.5

and for a fake coin,

P(

)=1

. You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is __________. (rounded off to two decimal places)

Consider a probability distribution given by the density function

P(x)

P(x)=\left\{\begin{aligned} C x^{2}, & \text{ for } 1 \leq x \leq 4 \\ 0, & \text{ for } x<1 \text{ or } x>4 \end{aligned}\right.

The probability that

x

2

3

P(2 \leq x \leq 3)

is _________. (rounded off to three decimal …

A quadratic polynomial

(x-\alpha)(x-\beta)

over complex numbers is said to be square invariant if

(x-\alpha)(x-\beta)=\left(x-\alpha^{2}\right)\left(x-\beta^{2}\right)

. Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are equal is __________. (rounded …

The unit interval

(0,1)

is divided at a point chosen uniformly distributed over

(0,1)

\mathbb{R}

into two disjoint subintervals. The expected length of the subinterval that contains

0.4

is __________. (rounded off to two decimal places)

Consider a permutation sampled uniformly at random from the set of all permutations of

\{1,2,3, \cdots, n\}

n \geq 4

X

be the event that

1

2

in the permutation, and

Y

3

4

. Which one of the following statements is TRUE?

A

B

be two events in a probability space with

P(A)=0.3, P(B)=0.5

P(A \cap B)=0.1

. Which of the following statements is/are TRUE?

When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers

(i.e., 1, 2, 3, 4, 5, \text{and } 6)

x

y

be random variables, not necessarily independent, that take real values in the interval

[0,1]

z=x y

and let the mean values of

x, y, z

\bar{x}, \bar{y}, \bar{z}

, respectively. Which one of the following statements is TRUE?

Consider a random experiment where two fair coins are tossed. Let

A

be the event that denotes

\text{HEAD}

on both the throws,

B

be the event that denotes

\text{HEAD}

on the first throw, and

C

be the event that denotes

\text{HEAD}

on the second throw. Which of the following statements is/are

\text{TRUE}?

The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter

2

. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to

2

decimal places) is ____________.

Consider the two statements.

S_1:\quad

There exist random variables

X

Y

\left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2>\textsf{Var}[X]\textsf{Var}[Y]

S_2:\quad

For all random variables

X

Y, \textsf{Cov}[X,Y]=\mathbb E \left[|X-\mathbb E[X]|\;|Y-\mathbb E[Y]|\right ]

Which one of the following …

For a given biased coin, the probability that the outcome of a toss is a head is

0.4

. This coin is tossed

1,000

X

denote the random variable whose value is the number of times that head appeared in these

1,000

tosses. The standard deviation of

X

2

decimal place) is _________

In an examination, a student can choose the order in which two questions (

\textsf{QuesA}

\textsf{QuesB}

) must be attempted. If the first question is answered wrong, the student gets zero marks. If the first question is answered correctly and the second question is not answered correctly, the student gets the marks only for the first question. If …

n>2

a \in \{0,1\}^n

be a non-zero vector. Suppose that

x

is chosen uniformly at random from

\{0,1\}^n

. Then, the probability that

\displaystyle{} \Sigma_{i=1}^n a_i x_i

is an odd number is______________

Y

is distributed uniformly in the open interval

(1,6)

. The probability that the polynomial

3x^2 +6xY+3Y+6

has only real roots is (rounded off to

1

decimal place) _______

P

Q

, decide to independently roll two identical dice, each with

6

faces, numbered

1

6

. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by

P

Q.

Assume that all

6

numbers on each dice are equi-probable and …

X

be a Gaussian random variable with mean 0 and variance

\sigma ^{2}

Y

\max\left ( X,0 \right )

\max\left ( a,b \right )

is the maximum of

a

b

. The median of

Y

is ______________ .