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

If a random variable

X

has a Poisson distribution with mean

5

, then the expectation

E\left [ \left ( x+2 \right )^{2} \right ]

For any discrete random variable

X

, with probability mass function

P(X=j)=p_j, p_j \geq 0, j \in \{0, \dots , N \}

\Sigma_{j=0}^N \: p_j =1

, define the polynomial function

g_x(z) = \Sigma_{j=0}^N \: p_j \: z^j

. For a certain discrete random variable

Y

, there exists a scalar

\beta \in [0,1]

Consider the following experiment. Step 1. Flip a fair coin twice. Step 2. If the outcomes are (TAILS, HEADS) then output

Y

and stop. Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output

N

and stop. Step 4. If the outcomes are (TAILS, TAILS), then go to Step

1.

The probability that the output of the experiment is

Y

A probability density function on the interval

[a, 1]

1/x^{2}

and outside this interval the value of the function is zero. The value of

a

X_i

i=1, 2, 3

are independent and identically distributed random variables whose probability mass functions are

Pr[X_i = 0] = Pr[X_i = 1] = \frac{1} {2} \text{ for } i = 1, 2, 3

. Define another random variable

Y = X_1X_2 \oplus X_3

\oplus

denotes XOR. Then

Pr[Y=0 \mid X_3 = 0] =

Four fair six-sided dice are rolled. The probability that the sum of the results being

22

\dfrac{X}{1296}

X

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .

The probability that a given positive integer lying between

1

100

(both inclusive) is NOT divisible by

2

3

5

Each of the nine words in the sentence

\text{"The quick brown fox jumps over the lazy dog”}

is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The

\text{expected}

length of the word drawn is _____________. (The answer should be rounded to one decimal place.)

The security system at an IT office is composed of

10

computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system …

S

be a sample space and two mutually exclusive events

A

B

A \cup B = S

P(.)

denotes the probability of the event, the maximum value of

P(A)P(B)

Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is

\dfrac{1}{2}.

What is the expected number of unordered cycles of length three?

p

is the number of cars per minute passing through a certain road junction between

5

6

p

has a Poisson distribution with mean

3

. What is the probability of observing fewer than

3

cars during any given minute in this interval?

Consider a random variable

X

that takes values

+1

−1

with probability

0.5

each. The values of the cumulative distribution function

F(x)

x = −1

+1

5

cards (each carrying a distinct number from

1

5

) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card?

Consider a finite sequence of random values

X=[x_1,x_2,\dots x_n]

\mu_x

be the mean and

\sigma_x

be the standard deviation of

X

. Let another finite sequence

Y

of equal length be derived from this as

y_i=a*x_i+b

a

b

are positive constants. Let

\mu_y

be the mean and

\sigma_y

be the standard deviation of …

If the difference between the expectation of the square of a random variable

\left(E\left[X^2\right]\right)

and the square of the expectation of the random variable

\left(E\left[X\right]\right)^2

R

What is the probability that divisor of

10^{99}

is a multiple of

10^{96}

Consider a company that assembles computers. The probability of a faulty assembly of any computer is

p

. The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of

q

. What is the probability of a computer being declared faulty?

X

be a random variable following normal distribution with mean

+1

4

Y

be another normal variable with mean

-1

and variance unknown. If

P (X \leq -1) = P (Y \geq 2)

, the standard deviation of

Y