GATE EC Probability and Statistics questions

Practise Probability and Statistics questions for GATE EC. 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

X, N, Y

Z

be random variables. The variables

X

N

are independent of each other.

X

is uniformly distributed between

-1

1 ; N

follows Normal distribution with zero mean and unity variance.

Y

Z

are defined as,

Y=X+N

Z=X^{2}+N

. Which of the following pairs represents the values of correlation …

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

A source transmits symbol

S

that takes values uniformly at random from the set

\{-2,0,2\}

. The receiver obtains

Y=S+N

N

is a zero-mean Gaussian random variable independent of

S

. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol

S

. Suppose the probability of symbol estimation error

P_{e}

The random variable

X

takes values in

\{-1,0,1\}

with probabilities

P(X=-1)=P(X=1)=\alpha

P(X=0)=1-2 \alpha

0<\alpha<1 / 2

g(\alpha)

denote the entropy of

X

(in bits), parameterized by

\alpha

. Which of the following statements is/are TRUE?

Two fair dice (with faces labeled

1,2,3,4,5

6

) are rolled. Let the random variable

X

denote the sum of the outcomes obtained. The expectation of

X

is __________ (rounded off to two decimal places).

\text{X}

\text{Y}

are independent and identically distributed random variables that are distributed uniformly in the interval

[0,1]

. The probability that

\text{X} \geq \text{Y}

\_\_\_\_\_\_

A random variable

\mathrm{X}

, distributed normally as

\mathrm{N(0,1)}

undergoes the transformation

\mathrm{Y}=\mathrm{h}(\mathrm{X})

, given in the figure. The form of the probability density function of

\mathrm{Y}

is (In the options given below,

a, b, c

are non-zero constants and

g(y)

is piece-wise continuous function).

H(X)

denote the entropy of a discrete random variable

X

K

possible distinct real values. Which of the following statements is/are necessarily true?

A state transition diagram with states

A, B,

C,

and transition probabilities

p_{1}, p_{2}, \dots, p_{7}

is shown in the figure (e.g.,

\text{p}_{1}

denotes the probability of transition from state

A

B

). For this state diagram, select the statement(s) which is/are universally true.

Two continuous random variables

X

Y

Y=2X+3

\sigma ^{2}_{X}

\sigma ^{2}_{Y}

denote the variances of

X

Y

, respectively. The variances are related as

A box contains the following three coins. A fair coin with head on one face and tail on the other face. A coin with heads on both the faces. A coin with tails on both the faces. A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the …

The random variable

Y=\int_{-\infty }^{\infty }W\left ( t \right )\phi \left ( t \right )dt, \text{ where } \phi \left ( t \right )=\begin{cases} 1; & 5\leq t\leq 7 &\\ 0; & \text{otherwise} \end{cases}

W(t)

is a real white Gaussian noise process with two-sided power spectral density

S_{W}\left ( f \right )=3 W/Hz

f

The two sides of a fair coin are labelled as

0

1

. The coin is tossed two times independently. Let

M

N

denote the labels corresponding to the outcomes of those tosses. For a random variable

X

X = \text{min}(M, N)

, the expected value

E(X)

(rounded off to two decimal places) is ___________.

X

is a random variable with uniform probability density function in the interval

[-2,\:10]

Y=2X-6

, the conditional probability

P\left ( Y\leq 7\mid X\geq 5 \right )

(rounded off to three decimal places) is __________.

X

Y

are random variables such that

E\left[2X+Y\right]=0

E\left[X+2Y\right]=33

E\left[X\right]+E\left[Y\right]=

Z

be an exponential random variable with mean

1

. That is, the cumulative distribution function of

Z

F_{Z}(x)= \left\{\begin{matrix} 1-e^{-x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.

Pr\left(Z>2 \mid Z>1\right),

rounded off to two decimal places, is equal to ___________.

A random variable

X

-1

+1

with probabilities

0.2

0.8

, respectively. It is transmitted across a channel which adds noise

N,

so that the random variable at the channel output is

Y=X+N

N

is independent of

X,

and is uniformly distributed over the interval

[-2,2].

The receiver makes a …

X_{1},\:X_{2},\:X_{3}

X_{4}

be independent normal random variable with zero mean and unit variance. The probability that

X_{4}

is the smallest among the four is ________.

A random variable

X

-0.5

0.5

with probabilities

\dfrac{1}{4}

\dfrac{3}{4}

, respectively. The noisy observation of

X\:\text{is}\:Y=X+Z,

Z

has uniform probability density over the interval

(-1,1).\: X

Z

are independent. If the MAP rule based detector outputs

\hat{X}

Three fair cubical dice are thrown simultaneously . The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place)________.