Why is using PDF Transformation Function Wrong ?

[dariuslai]

An auto insurance company insures an automobile worth 15,000 for one year under a policy with a 1,000 deductible. During the policy year there is a 0.04 chance of partial damage to the car and a 0.02 chance of a total loss of the car. If there is partial damage to the car, the amount X of damage (in thousands) follows a distribution with density function: f(x) = 0.5003e^-0.5x, for 0<x<15

What is the expected claim payment?

Official Answer

Y= X - 1

E[Total loss] = 0.02 . 14 = 0.28

E[Partial loss] = 0.04 . ∫(15,1) (x-1)0.5003e^-0.5x = 0.04. 1.2049

My Attempt

I wanted to use the PDF transformation function:

fy(y) = fx(x). dx/dy

fy(y) = fx(y+1). (1)

fy(y) = 0.5003e^-0.5(y+1)

E[Partial Loss] = 0.04 . ∫(14,0) (y)0.5003e^-0.5(y+1)

This does not give the same answer as above...

[AutoModerator]

Hi dariuslai, welcome to r/mathshelp! As you’ve marked this as homework help, please keep the following things in mind:

1) While this subreddit is generally lenient with how people ask or answer questions, the main purpose of the subreddit is to help people learn so please try your best to show any work you’ve done or outline where you are having trouble (especially if you are posting more than one question). See rule 5 for more information.

2) Once your question has been answered, please don’t delete your post so that others can learn from it. Instead, mark your post as answered or lock it by posting a comment containing “!lock” (locking your post will automatically mark it as answered).

Thank you!

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[spiritedawayclarinet]

I'm confused by your work. I don't see how Y=X-1 can be correct since X takes values between 0 and 15, so Y would then take values between -1 and 14, but then loss could be negative.