Question

# The measurement error in estimating the fortune of a certain self-declared billionaire, X, has the following...

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?

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

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