Question

The measurement error in estimating the fortune of a certain self-declared billionaire, X, has the following cumulative distribution function:

F(x) = 0 for x < -2
F(x)= .5 + .09375(4x −x³/3) for -2 ≤ x ≤ 2

F(x) = 1 for x > 2

a) Give the probability density function for X in the interval -2, 2].

b) What is the probability that X > 1?  

c) What is the probability that the billionaire actually has a fortune less than -2?  

d) What is the probability that X > .5?  

e) What is the probability that X > 1 given that X > .5?  

f) Calculate the 37th percentile of X.  

g) What is the expected value of X?  

h) What is the expected value of X²?  

i) What is the variance of X?  

j) What is the probability that X is more than 1 above its expected value?

Answer 1

a)

pdf f(x)=(d/dx)F(x)=0.09375*(4-3x²)

b)P(X>1) =1-P(X<1)=1-(0.5+0.09375*(4*1-1³/3))=0.15625

c)P(X<-2)=0

d)P(X>0.5)=1-P(X<0.5)=1-(0.5+0.09375*(4*0.5-0.5³/3))=0.316406

e)P(X>1|X>0.5)=P(X>1)/P(X>0.5)=0.15625/0.316406=0.493827

f)for 37 th percentile ; x=-0.35

g)E(X)= xf(x) dx = 0.09375*(4x-x³) dx =0.09375*(4x²/2-x⁴/4)|²_-2 =0

h)

E(X²)= x²f(x) dx = 0.09375*(4x²-x⁴) dx =0.09375*(4x³/3-x⁵/5)|²_-2 =0.8

i)

Var(X)=E(X²)-(E(X))² =0.8

j)

P(X>1+0)=P(X>1)=0.15625