Question: What is the variance of a continuous random variable Y with probability density function (pdf) given by g(y) = 4y^3, 0 < y < 1?

Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.1: Continuous Probability Models
Problem 25E
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Question: What is the variance of a continuous random variable Y with probability density function (pdf) given by g(y) = 4y^3, 0 < y < 1?
Expert Solution
Step 1

The variance of Y, denoted by Var(Y), is defined as the expected value of the squared deviation from the mean and can be calculated as follows:

Var(Y) = E((Y - E(Y))^2) = E(Y^2) - (E(Y))^2

= ∫y^2 * g(y) dy - (E(Y))^2

= ∫y^2 * 4y^3 dy - (E(Y))^2

= ∫4y^5 dy - (E(Y))^2

= y^6/6 | from 0 to 1

= (1^6)/6 - (0^6)/6

= 1/6 - 0

= 1/6

So the variance of Y is 1/6.

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