---
title: "Ch 4.1 Practice: PDFs and CDFs"
author: "Name"
output: pdf_document
---

# 1. Let $X$ be a random variable with pdf given by $f(X)=2x$ for $0\leq x\leq 1$ and 0 otherwise.

## a. Find $P(X\geq 1/2)$


## b. Find $P(X\geq1/2|X\geq 1/4)$.



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# 2. Let $X$ be a random variable with pdf given below. Find the constant c such that this is a valid pdf. 

$$
f(x)= \Bigg\{
\begin{array}{cc}
cx^{2} & \mbox{for }0 \leq x < 1 \\
c(2-x)^{2} & \mbox{for }1 \leq x \leq 2 \\
0  & otherwise.
\end{array}
$$

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# 3. Suppose that the p.d.f. of a random variable $X$ is as follows:

$$
f(x)= \Big\{ \begin{array}{cc} 
\frac{1}{8}x & for\ 0\leq x \leq 4 \\
0 & otherwise.
\end{array}
$$

## a. Find the value of $t$ such that P$(X\leq t)=1/4$.


## b. Find the value of $t$ such that P($X\geq t)=1/2$.


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# 4. Suppose $f_{Y}(y)=4y^{3}$, $0\leq y\leq 1$.  Find $P(0\leq Y\leq \frac{1}{2})$



\newpage
# 5. Let $f(y)=\frac{3}{2}y^{2}$ for $-1\leq y\leq 1$. Find $P(|Y-\frac{1}{2}|<\frac{1}{4})$.



\newpage
# 6. The cdf for a random variable $Y$ is defined by $F(y)=0$ for $y<0$; $F(y)=4y^{3}-3y^{4}$ for $0\leq y\leq 1$; and $F(y)=1$ for $y>1$. Find $P(\frac{1}{4}\leq Y\leq \frac{3}{4})$.


