Chapters: Expected value & Variance of a continuous random variable **Q1.** A box is to be constructed so that its height is five inches and its base is $ Y $ inches by $ Y $ inches, where $ Y $ is a random variable described by the probability density function (pdf), $ f(y) = 6y(1-y) $, $ 0 < y < 1 $. Find the expected value of the volume of the box.---**Q2.** Let $ X $ be a random variable with the probability density function $ f(x) = 3(1-x)^2 $ when $ 0 \leq x \leq 1 $ and $ f(x) = 0 $ otherwise.**a.** Verify that $ f $ is a valid pdf.**b.** Find the mean and variance of $ X $.**c.** Find $ P(X \leq 1/2) $.**d.** Find $ P(X \leq 1/2|X \geq 1/4) $.

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Q1. A box is to be constructed so that its height is five inches and its base is Y inches by Y inches, where Y is a random variable described by the pdf, f(y) = 6y(1 – y), 0 < y < 1. Find the expected value of the volume of the box.

Q1. A box is to be constructed so that its height is five inches and its base is Y inches by Y inches, where Y is a random variable described by the pdf, f(y) = 6y(1 – y), 0 < y < 1. Find the expected value of the volume of the box.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Chapters: Expected value & Variance of a continuous random variable

**Q1.** A box is to be constructed so that its height is five inches and its base is $ Y $ inches by $ Y $ inches, where $ Y $ is a random variable described by the probability density function (pdf), $ f(y) = 6y(1-y) $, $ 0 < y < 1 $. Find the expected value of the volume of the box.

---

**Q2.** Let $ X $ be a random variable with the probability density function $ f(x) = 3(1-x)^2 $ when $ 0 \leq x \leq 1 $ and $ f(x) = 0 $ otherwise.

**a.** Verify that $ f $ is a valid pdf.

**b.** Find the mean and variance of $ X $.

**c.** Find $ P(X \leq 1/2) $.

**d.** Find $ P(X \leq 1/2|X \geq 1/4) $.

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