An insurance company allows various time between payments in

An insurance company allows various time between payments (in months). A random variable can be defined as X = number of months between two payments.

The CDF of X =

a) What is the PDF?

b) What is the variance V[X]?

c) Compute the values of P(3<= X < 12)

Solution

a) What is the PDF?

P(X=1) =0.1

P(X=2)=0.25-0.1 =0.15

P(X=3)=0.5-0.25=0.25

P(X=5)=0.55-0.5=0.05

P(X=7)=0.75-0.55=0.2

P(X=12)=1-0.75=0.25

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b) What is the variance V[X]?

E(X)=sum of x*f(x)

=1*0.1+2*0.15+3*0.25+5*0.05+7*0.2+12*0.25 =5.8

E(X^2)=sum of x^2*f(x)

=1*0.1+2^2*0.15+3^2*0.25+5^2*0.05+7^2*0.2+12^2*0.25 =50

SO V(X)= E(X^2) -[E(X)]^2

=50-5.8^2

=16.36

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c) Compute the values of P(3<= X < 12)

=P(X=3)+P(X=5)+P(X=7)

=0.25+0.05+0.2

=0.5