JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (December - 2018 - R16)

State theorem of total probability.

Explain the terms events and mutually exclusive events with examples.

Define skew of a random variable.

Write the properties of probability density function.

Write the expressions to obtain marginal distribution functions from joint density function.

State central limit theorem for unequal distributions.

Explain strict sense stationarity of a random process.

Write short notes on Poisson random process.

Write Wiener- Khintchine relations.

Is the function represented a valid power spectral density? Mention properties used.

State and prove Bayes theorem.

Discuss relative frequency approach of probability.

A pointer is spun on a fair wheel of chance having its periphery labeled between 1 and 100. i) What is the probability that the pointer will stop between 20 and 35? ii) What is the probability that the wheel will stop on 58.

Classify random variables and explain.

Find the mean of Poisson random variable.

A random variable X is uniformly distributed in the interval (-2,2). It is transformed to a new variable Y by the transformation Y = CX². Obtain the density of Y.

In a sports event javelin throw distances are well approximated by a Gaussian distribution for which mean is 30m and standard deviation is 5m. In a qualifying round, contestants must throw farther than 26m to qualify. In the main event the record throw is 42m. i) What is the probability of being disqualified in the first round? ii) In the main event what is the probability the record will be broken?

Obtain the characteristic function for Binomial random variable.

Derive the expression for conditional density considering interval conditioning.

The random variables Y₁, Y₂, X₁, X₂ are related by the relations: Y₁ = aX₁ - bX₂ and Y₂ = cX₁ - dX₂. Find f_Y1,Y2(y₁,y₂) in terms of joint density of X₁ and X₂.

Check for the validity of joint distribution G_X,Y(x,y) = u(x)u(y)[1 - e^-(x+y)]. Mention the properties used.

Prove that the mean of weighted sum of random variables is equal to the weighted sum of means of the individual random variables.

State and prove any three properties of Auto correlation function of a random process.

A random process X(t) = A Cos(ω₀t + Θ) where A and ω₀ are constants and Θ is a uniformly distributed random variable in (0,π), check whether random process is WSS or not?

X(t) is W.S.S. random process and X(t) = A Cos(w₀t + Θ). Given A is a uniformly distributed random variable in the interval (-a, b). Find E[X(t)], R_XX(t,t+τ).

Define cross correlation function and write its properties.