---
title: "Ch 4.2 & 4.3 Practice: Expected value & Variance of a continuous random variable"
author: "YOUR NAME HERE"
output: pdf_document
---


# 1. Calculate $E(Y)$ for the following pdfs:
## a. $f(y)=3(1-y)^{2}$, $0\leq y\leq 1$



## b. $f(y)=4ye^{-2y}$, $y\geq 0$




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# 2. If $Y$ has pdf $f(y)=2y$ for $0<y<1$. Then $E(Y)=\frac{2}{3}$. Define the random variable $W$ to be squared deviation of $Y$ from its mean, that is, $W=(Y-\frac{2}{3})^{2}$. Find $E(W)$. 



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# 3. 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.



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# 4. Let $X$ be a random variable with pdf $f(x)=3(1-x)^{2}$ 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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# 5. Let $f(y)=3(1-y)^{2}$ for $0<y<1$. Find the variance of $W$ where $W=-5Y+12$.



