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Chapter 4: Q97E (page 262)

Variable life insurance return rates. With a variable life insurance policy, the rate of return on the investment (i.e., the death benefit) varies from year to year. A study of these variable return rates was published in International Journal of Statistical Distributions (Vol. 1, 2015). A transformedratio of the return rates (x) for two consecutive years was shown to have a normal distribution, with 渭=1.5 and role="math" localid="1660283206727" 蟽=0.2. Use the standard normal table or statistical software to find the following probabilities.

a.P(1.3<x<1.6)

b. P(x>1.4)

c. P(x<1.5)

Short Answer

Expert verified
  1. P1.3<x<1.6=0.5328
  2. role="math" localid="1660283402755" Px>1.4=0.6915

c. Px<1.5=0.50

Step by step solution

01

Given information

Variable life insurance return rates followa normal distribution with渭=1.5 and蟽=0.2

02

Finding the z values

z=x-渭蟽=x-1.50.2

For, x = 1.6

Then,

z=x-渭蟽=1.6-1.50.2=0.5

For, x=1.3

Then,

z=x-渭蟽=1.3-1.50.2=-1

For, x=1.4

Then,

z=x-渭蟽=1.4-1.50.2=-0.5

For, x=1.5

Then,

z=x-渭蟽=1.5-1.50.2=0

03

Probability calculation when P(1.3<x<1.6)

a.

P1.3<x<1.6=Px<1.6-Px<1.3=Pz<0.5-Pz<-1=Pz<0.5-1-Pz<-1=0.6915-1-0.8413=0.6915-1+0.8413=0.5328P1.3<x<1.6=0.5328

Therefore, the value of P(1.3<x<1.6) is 0.5328.

04

Probability calculation when P(x>1.4)

Px>1.4=1-Px<1.4=1-Pz<-0.5=1-1-Pz<-0.5=0.6915Px>1.4=0.6915

Therefore, the value ofPx>1.4 is 0.6915.

05

Probability calculation when P(x<1.5)

b.

P(x<1.5)=P(z<0)=0.50P(x<1.5)=0.50

Therefore, the value of P(x<1.5) is 0.50.

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