JNTU B.Tech II Year PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (May/June - 2019 - R15)

Three machines A, B and C produce 55%, 25% and 20% of the total number of items of a factory. The percentage of defective output of these machines are 3%, 2% and 4%. If an item is selected at random, (i) find the probability that the item is defective (ii) If the selected item is defective, find the probability that the item is produced by Machine B.

In a single throw of a pair of dice, what is the probability of obtaining the sum of two faces of the dice is equal to at least 10.

Two different digits are selected at random from the digits 1 to 9. (i) If the sum of the digits is odd, what is the probability that 2 is one of the digits selected? (ii) If 2 is one of the selected digits, what is the probability for the sum to odd.

A binary communication system transmits a 0 and 1 with equal probabilities. Due to the noise in the channel, a transmitted 1 is received as a 0 with a probability of 1/8 and a transmitted 0 is received as 1 with a probability of 3/4. (i) Find the probability for the transmitted message to be a 1. (ii) If a one is received, find the probability that the transmitted is a 1.

A discrete random variable is having a density function given by f(x)=0.1δ(x)+0.4δ(x-1)+0.3δ(x-2)+0.2 δ(x-3). Find the density of the random variable Y = -x²+2x+3.

Verify that Gaussian Density function is a valid density function.

Find the Mean and Mean square value of Rayleigh Random Variable.

X is a Gaussian random variable with zero mean and unity variance. Find the density of the random variable Y = X for x ≥ 0, and Y=0 for x < 0.

Two random variables X and Y have a joint probability density function f(x,y)= (5/16)x²y for 0 < y < x < 2; and zero elsewhere. Check X and Y for independence.

Let f(x,y) = x + y for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1; and zero elsewhere. Find the density of X, given Y.

The joint probability mass function of two random variables X and Y is P(x,y)=K(2x+3y) for x=0,1,2; and y=1,2,3. Find the marginal probability distributions of X and Y. Find all the possible conditional probabilities of X.

If X is a random variable with mean 3 and variance 2, verify that the random variables X and Y=-6X+22 are orthogonal.

A Random Process X(t)=A.Cos (2πfct) , where A is a Gaussian Random Variable with zero mean and unity variance, is applied to an ideal integrator, that integrates with respect to t, over (0,t). Check the output of the integrator for stationarity.

In the random Process X(t)=A.CosWt, A is uniform random Variable over (0,1) and W is a constant. Find the Auto correlation function of X(t).

X(t) is a random process with mean =3 and Autocorrelation function Rxx(τ) =10.[exp(-0.3|τ|)+2]. Find the second central Moment of the random variable Y=X(3)-X(5).

X(t) is a WSS process and Y(t)=A.Cos(Wct+θ) is a random process which is independent of X(t). Here, θ is a uniform random variable over(-π,π). If the Auto correlation function of X(t) is Rxx(τ), find the Auto correlation of Z(t)=X(t).Y(t).

Find the Auto correlation and PSD and M.S. Value of the random process, X(t)= m(t). Cos(Wt+Ф), where m(t) is a WSS process and Ф is a uniform random variable over (0,2π), and is independent of m(t).

A noise process with zero mean and of PSD K is applied to an R-L LPF. Find the Mean Square value of the output Process.

Let x(t)=Y(t) -Y(t-2) is a random process, where Y(t) is also a stationary random process. It is given Var(X(t))=20.Var(Y(t)). Find RYY(2)/var(X(t)) and also BXX.

X(t)=A.Sin(wt+θ) is a random process, with θ being a uniform random variable over the interval (-π, π). If Y(t)=(1/2)X(t). Are X(t) and Y(t) are jointly stationary and find Syy.