JNTU B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (November/December - 2018 - R15)

The joint occurrence of two events A and B is given as a matrix. Find the probability of B is even, given that A is even.

A die is weighted so that even numbers have same chance of appearing, and the odd numbers have the same chance of appearing, and each even number is twice as likely to appear as any odd number. When the die is tossed, what is the probability of getting an even numbered face.

A random variable X has a CDF given by F(x) = (1 - cos(x))/2 for 0 ≤ x ≤ π. Find F(x) at x = π/2, and justify the answer.

A random variable X has the following distribution: xi values 0-8 with probabilities a, 3a, 5a, 7a, 9a, 11a, 13a, 15a, 17a. Find the smallest value of x, for which P[(X ≤ x) > 0.5].

A fair coin is tossed three times. Let X denote a ‘0’ or a ‘1’ according a head or tail occurs in first toss and let Y denote the number of heads which occur. Find the joint distribution of X and Y.

Find the correlation coefficient between random variables X and Y related as X=20Y.

A random process is defined as X(t)= A.Cos(ωt), where A is a uniform random variable. What is the condition on A for the process to be WSS?

X(t)=A Cos(2πt+Y) is a random process, where Y is a random variable such that P(Y=0)=1/2 and P(Y=π/2)= 1/2. Find the correlation between the random variables X(0) and X(1).

A random process with PSD of K watts/Hz is applied to an ideal LPF with pass band over (-B Hz to +B Hz). Find the noise power at the output of the filter.

Find the cross spectral density of two uncorrelated stationary random processes X(t) and Y(t).

A coin is flipped three times and X denotes the number of heads that show up. The probability of getting a head in each flip is ‘q’. Give the probability distribution of X. Find the probability with which X>1.

A pair of fair dice is thrown. Find the probability that the sum is 10 or greater if (i) 5 appears on the first dice (ii) 5 appears at least on one of the dice.

Find the probability of getting a total of 4 at least once in 3 tosses of a pair of dice.

Two different digits are selected at random from digits 1 to 9. If the sum of the digits selected is odd, what is the probability for 2 to be one of the numbers selected?

A pair of fair dice is tossed. Find the probability that the maximum of the two numbers is greater than four.

The pdf of a random variable X is given. Find and plot the CDF of X.

The density function of a random variable X is given. If three independent observations are made on X, find the probability with which exactly one of them is greater than 1.5.

Find the density of the Random variable Y=Tan(X), where X is uniform random variable over (-π/2, π/2).

For a random variable X with pdf f(x)= 0.5 exp(-|x|), find the probability that an observed value will fall within ±2σ^2 of the mean value.

Find the density of Z=X+Y, where X and Y are two independent random variables, which are uniform over (-2, +2) and (-1, 3) respectively.

A product is classified according to the number of defects it contains (X) and the factory (Y) producing the product. Find the conditional distribution of X, given Y=1 and check if X and Y are independent.

The input to a binary communication channel is a random variable X. Given specific conditional probabilities, find the probability of receiving 0 and the conditional probability of transmitted 1 given received 0.

If X and Y are two independent variables, verify that their sum and difference are of same variance.

Let X(t) be a random process with mean 3 and auto correlation 9+4.exp(-0.2|τ|). Find the mean, variance and covariance of the random variables X(5) and X(8).

Check random process X(t)=A.cos(ωt+β), where β is a uniform random variable over (0, 2π) for mean ergodicity.

A random process is defined as X(t)=ACos(100t+β), where A is a normal random variable. Find the Autocorrelation function of X(t).

Which of the following are valid Autocorrelation functions? Justify: (i) A.Cosωτ (ii) A.Sinωτ (iii) A[u(t+τ)-u(t-τ)] (iv) Triangular pulse.

A random process with psd of K watts/Hz is applied to an RC LPF with 3dB cutoff frequency of fc. Find the power at the output of the filter.

A random process X(t) has an autocorrelation function. Find the mean of the output of a system with unit impulse response exp(-kt).u(t).

Two systems with identical unit impulse response of t.exp(- kt)u(t) are in cascade. If the cascade is driven by a WSS process with a mean of 2, find the mean of the output of the cascade.

A random process X(t) with Autocorrelation function is applied to an LTI system with unit impulse response of K.exp(-Kt).u(t). Find the Autocorrelation of the response of the system.