I (Random Variables) - Hariganesh.com

Engineering Mathematics

2016

SUBJECT NAME

: Probability & Random Process

SUBJECT CODE

: MA6451

MATERIAL NAME

: Part – A questions

MATERIAL CODE

: JM08AM1008

REGULATION

: R2013

UPDATED ON

: February 2016

(Scan the above Q.R code for the direct download of this material)

Name of the Student:

Branch:

Unit – I (Random Variables) 1) Define random variable. 2) Define discrete random variable. 3) Define continuous random variable. 4) X and Y are independent random variables with variance 2 and 3. Find the variance of 3 X + 4Y .

x in 0 ≤ x ≤ 2 , find P ( X > 1.5 / X > 1) . 2 6) Assume that X is a continuous random variable with the probability density 5) If the p.d.f of a random variable X is f ( x ) =

3 2  (2x − x ) , 0 < x < 2 function f ( x ) =  4 . Find P ( X > 1) .  0, otherwise 7) A continuous random variable X has probability density function

3 x 2 , 0 ≤ x ≤ 1 f ( x) =  . Find k such that P ( X > k ) = 0.5 . otherwise 0, 8) Find c , if a continuous random variable X has the density function c f ( x) = , −∞ < x < ∞. 1 + x2 1 9) If the MGF of a uniform distribution for a random variable X is ( e 5 t − e 4 t ) , find E ( X ) . t 10) The moment generating function of a random variable X is given by M ( t ) = e

(

3 e t −1

) . What

is P [ X = 0] ?

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

Page 1


2016 x 0, y > 0 . Find the

value of k . 15) The two regression equations of two random variables X and Y are

4 x − 5 y + 33 = 0 and 20 x − 9 y = 107 . Find the mean values of X and Y . 16) The regression equations are 3 x + 2 y = 26 and 6 x + y = 31 . Find the correlation coefficient between X and Y .


Page 3


2016

Unit – III (Classification of Random Processes) 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)

Define random process. Define stochastic processes. Define a wide sense stationary process. Define a strictly stationary random process (SSS). Define Markov process Define a Markov chain and give an example. Prove that a first order stationary process has a constant mean. Explain any two applications of a binomial process. Show that a binomial process is Markov. State the postulates of a Poisson process. Write down any two properties of Poisson process. Prove that sum of two independent Poisson processes is again a Poisson process. When is a random process said to be mean ergodic? State the properties of an ergodic process.

15) If { X ( t )} is a normal process with µ ( t ) = 10 and C ( t1 , t 2 ) = 16e

− t1 − t 2

find the variance of

X (10) − X (6) . 16) Consider the random process X ( t ) = cos( t + φ ) , where φ is a random variable with density function f (φ ) =

1

π

,

−π π < φ < . Check whether or not the process is wide sense 2 2

stationary.

Unit – IV (Correlation and Spectral densities) 1) Define power spectral density function. 2) Find the power spectral density function of the stationary process whose autocorrelation −τ

function is given by e . 3) Prove that the spectral density of a real random process is an even function. 4) The autocorrelation function of a stationary random process is R(τ ) = 16 +

9 . Find 1 + 16τ 2

the mean and variance of the process. 5) Find the mean and variance of the stationary ergodic process { x( t )} whose auto

4 . 1 + 6τ 2 6) Find the variance of the stationary process { x( t )} whose auto correlation function is given correlation function is given by R(τ ) = 25 +

by RXX (τ ) = 2 + 4e

−2 τ

.

7) Prove that for a WSS process { X ( t )} , RXX ( t , t + τ ) is an even function of τ .


Page 4


2016

8) Prove that S xy (ω ) = S yx ( −ω ) . 9) Define cross correlation function of X ( t ) and Y( t ) . When do you say that they are independent? 10) Write any two properties of autocorrelation. 11) State any two properties of cross correlation function. 12) A random process X ( t ) is defined by X ( t ) = Kcos ω t, t ≥ 0 where ω is a constant and

K is uniformly distributed over (0,2). Find the auto correlation function of X (t ) . 13) State Wiener-Khinchine theorem.

Unit – V (Linear systems with Random inputs) 1) 2) 3) 4) 5) 6) 7) 8)

Define a system. When is it called a linear system? Define casual system. Define a linear time – invariant system.

Define transfer function of a system. State the convolution form of the output of a linear time invariant system. Define Band-Limited white noise. State autocorrelation function of the white noise. Find the system Transfer function, if a Linear Time Invariant system has an impulse function

1 ;t ≤c  H ( t ) =  2c . 0 ; t ≥ c  9) Define white noise. 10) Prove that the system y ( t ) =

∞

∫ h( u) X (t − u)du is a linear time-invariant system.

−∞

11) What is unit impulse response of a system? Why is it called so? 12) If Y ( t ) is the output of an linear time invariant system with impulse response h( t ) , then find the cross correlation of the input function X ( t ) and output function Y ( t ) . 13) Sate any two properties of a linear time – invariant system. 14) If { X ( t )} and {Y ( t )} in the system Y ( t ) =

∞

∫ h( u) X (t − u) du are WSS process, how are

−∞

their auto correlation function related.

----All the Best---Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

Page 5