Question

SELF ASSESSMENT 2 An insurance company offers policyholders a number of different Premium payment options. For a randomly selected policyholder, let X be the number of months between successive payments. The cumulative distribution function,cdf, of X is as follows: F(x) = 0x< 1 0.30 15x<3 0.40 35x< 4 10.45 4 < x < 6 0.60 6<x< 12 x 12 1 i. Determine the probability distribution function, f(x). ii. Find the expectation and standard deviation of X. iii. Compute,P(3 SXS 6).

Answer 1

i) from given cdf ,

probability density function is:

f(1) =P(X<=1)-P(X<1) =0.3-0=0.3

f(3) =P(X<=3)-P(X<3) =0.4-0.3 =0.1

f(4) =P(X<=4)-P(X<4) =0.45-0.4 =0.05

f(6) =P(X<=6)-P(X<6)=0.6-0.45 =0.15

f(12) =P(X<=12)-P(X=12)=1-0.60 =0.40

ii)

from above

x	f(x)	xP(x)	x2P(x)
1	0.3000	0.3000	0.3000
3	0.1000	0.3000	0.9000
4	0.0500	0.2000	0.8000
6	0.1500	0.9000	5.4000
12	0.4000	4.8000	57.6000
	total	6.5000	65.0000
	E(x) =μ=	ΣxP(x) =	6.5000
	E(x2) =	Σx2P(x) =	65.0000
	Var(x)=σ2 =	E(x2)-(E(x))2=	22.7500
	std deviation=	σ= √σ2 =	4.7697

expectation E(x)=6.50

standard deviation SD(X) =4.7697

iii)

P(3<=X<=6) =f(3)+f(4)+f(6) =0.1+0.05+0.15 =0.30