GATE EC Probability and Statistics questions

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

19 on this page

SubjectTopicYearType

Two independent random variables

X

Y

are uniformly distributed in the interval

[-1,1]

. The probability that max

[X,Y]

\frac{1}{2}

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

A binary symmetric channel (BSC) has a transition probability of

\frac{1}{8}

. If the binary transmit symbol

X

P(X=0)\:=\:\frac{9}{10}

, then the probability of error for an optimum receiver will be

A probability density function is of the form

\qquad p(x)=\mathrm{Ke}^{-\alpha|x|}, x \in(-\infty, \infty)

\text{K}

Three companies

\text{X, Y}

\text{Z}

supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below …

A uniformly distributed random variable

X

with probability density function

f_{\gamma}(x)=\frac{1}{10}(t r(x+5)-u(x-5))

u(.)

is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random variable

Y

A fair dice is rolled twice. The probability that an odd number will follow an even number is

The value of the integral

\text{I}=\frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} \exp \left(-\frac{x^{2}}{8}\right)

d x

An output of a communication channel is a random variable

v

with the probability density function as shown in the figure The mean square value of

v

The distribution function

\mathrm{F}_{1}(x)

of a random variable

\mathrm{X}

is shown in the figure. The probability that

X=1

A random variable

\mathrm{X}

with uniform density in the interval

0

1

is quantized as follows:

\begin{array}{ll} \qquad \text{If 0 < X < 0.3,} & \text{x}_{4} = 0 \\ \qquad \text{If 0.3 < X < 1,} & \text{x}_{4} = 0.7 \end{array}

\text{x}_{4}

is the quantized value of

\text{X}

. The root-mean square value of …

X

Y

be two statistically independent random variables uniformly distributed in the ranges

(-1,1)

(-2,1)

respectively. Let

Z=X+Y

. Then the probability that

(Z \leq-2)

Data for Q. 65-66 are given below. Solve the problems and choose the correct answers. Let

X

be the Gaussian random variable obtained by sampling the process at

t=t_{i}

\mathrm{Q}(\alpha)=\int_{\alpha}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{\frac{x^{2}}{2}} d y

The probability that

[x \leq 1]

Y

Z

be the random variables obtained by sampling

X(t)

t=2

t=4

respectively. Let

W

=Y-Z

. The variance of

W

The PDF of a Gaussian random variable

X

\operatorname{Px}(x)=\frac{1}{3 \sqrt{2 \pi}} e^{\frac{-(x-4)^{2}}{18}}

The probability of the event

\{X=4\}

During transmission over a communication channel, bit errors occur independently with probability p. If a block of

n

bits is transmitted, the probability of at most one bit error is equal to

The probability density function of the envelope of narrow band Gaussian noise is

A probability density fucntion is given by

p(x)=\mathrm{K} e^{-x^2 / 2}-\infty<x<\infty

K

For a random variable

x

following the probability density function,

\mathrm{p}(x)

, shown in given figure the mean and the variance are, respectively,