Question 1200998: A continuous random variable Y has the pdf
\( fYes=C(2y+1), 0 < y < 2 . Find P(y<0.5)
Write answer in decimal form.
Answer by asinus(45)   (Show Source):
You can put this solution on YOUR website! **1. Determine the value of the constant C**
* For a valid probability density function (PDF), the total area under the curve must equal 1.
* Mathematically, this means:
∫₀² C(2y + 1) dy = 1
* Solve the integral:
C ∫₀² (2y + 1) dy = C [y² + y]₀² = C [(4 + 2) - (0 + 0)] = 6C
* Set this equal to 1:
6C = 1
C = 1/6
**2. Find P(Y < 0.5)**
* P(Y < 0.5) is the area under the PDF curve from 0 to 0.5.
* Calculate the integral:
∫₀⁰.⁵ (1/6)(2y + 1) dy = (1/6) ∫₀⁰.⁵ (2y + 1) dy
= (1/6) [y² + y]₀⁰.⁵
= (1/6) [(0.25 + 0.5) - (0 + 0)]
= (1/6) * 0.75
= 0.125
**Therefore, P(Y < 0.5) = 0.125**
|
|
|
