Solution Found!
If Y is a geometric random variable, define Y ? = Y ? 1.
Chapter 3, Problem 88E(choose chapter or problem)
Problem 88E
If Y is a geometric random variable, define Y ∗ = Y − 1. If Y is interpreted as the number of the trial on which the first success occurs, then Y ∗ can be interpreted as the number of failures before the first success. If Y ∗ = Y − 1, P(Y ∗ = y) = P(Y − 1 = y) = P(Y = y + 1) for y = 0, 1, 2, . . . . Show that
P ( Y ∗ = y) = q y p, y = 0, 1, 2, . . . .
The probability distribution of Y ∗ is sometimes used by actuaries as a model for the distribution of the number of insurance claims made in a specific time period.
Questions & Answers
QUESTION:
Problem 88E
If Y is a geometric random variable, define Y ∗ = Y − 1. If Y is interpreted as the number of the trial on which the first success occurs, then Y ∗ can be interpreted as the number of failures before the first success. If Y ∗ = Y − 1, P(Y ∗ = y) = P(Y − 1 = y) = P(Y = y + 1) for y = 0, 1, 2, . . . . Show that
P ( Y ∗ = y) = q y p, y = 0, 1, 2, . . . .
The probability distribution of Y ∗ is sometimes used by actuaries as a model for the distribution of the number of insurance claims made in a specific time period.
ANSWER:
Solution :
Step 1 of 1:
Given Y is a geometric random variable.
Let Y is interpreted as the number of the trial on which the first success occurs.
Let is interpreted as the number of failures before the first success.
Our goal is:
We need to show that P()=