The pdf of a random variable X is as shown below: (a) If the mean of X is m x =

Question

The pdf of a random variable X is as shown below:

(a) If the mean of X is mx = E[X ] =2 , then what is the value of c (give a numerical answer).

(c) Assuming that c = 6, compute the variance of X .

(d) Another random variable Y is defined as

Y = { 7, with probability of 3/4

={ 0, with probability of 1/4

sketch the pdf of Y.

(e) Find the mean and variance of Y .

(f) A new random variable Z is formed by adding up X and Y , that is,

Z = X + Y . Assuming that X and Y are independent random variables, find the

probability density function of Z . Choose c=6.

Explanation / Answer

a) c=4 . Mean is 2 Therefore c/2 =2.

b) Probability of getting a number upto 3 is 4/7 . Count from 0 to 3 divided by count from 0 to 6

c) 4.67

d) PDF sketch is same as above diagram but only 2 labels on X, 1/4 on 0 and 3/4 on 7 , rest all probabity are 0

e) Y has mean ( 0,7,7,7) = 5.35 and Var of 3.5.

f) Therefore, 0,1,2,2,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,,7,7,7. Prob densify function will be a frequenct distributoin also known as histogram

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