Question

1. Given PMF f(x) =(x^2−3)/20 for X= 2,3,4

•Make a probability distribution table for X

•Find the CDF of X,F(x)

•Find the mean of X,μX, and the S.D. of X,σX

•If Y=1/3 X+ 2, find the mean of Y,μY, and the S.D. of YσY

Answer 1

1) Given PMF of X,

for x = 2, 3 , 4.

a) Probability distribution table for X

X	2	3	4
P(X=x)	1/20	3/10	13/20

b) The CDF of X is given by,

   1/20 for x = 2

= (1+6)/20 = 7/20 for x <=3

= (1+6+13)/20 = 20/20 = 1 for x <=4.

c) The mean of X is given by,

= 2*(1/20) + 3* (3/10) + 4* (13/20)

= 3.6 (ans)

The standard deviation of X is given by,

  

= 2² * (1/20) + 3² * (3/10) + 4² * (13/20)

= 13.3

   = 13.3 - (3.6)² = 0.34

= 0.58 (ans)

d) Given ,

the mean of Y, E(Y) = 1/3 * E(X) + 2 (as E(aX) = a E(X) and E(a) = a)

= 3.6/3 + 2 = 3.2 (ans)

the Variance of Y, V(Y) = 1/9 V(X) (as V(aX) = a² V(X) and V(a) = 0)

= 0.34/9 = 0.0378

then standard deviation of Y is

= 0.194 (ans)

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