## 4.4 End of Chapter Problems

Problem

Choose a real number uniformly at random in the interval $[2,6]$ and call it $X$.

1. Find the CDF of X, $F_X (x)$.
2. Find $EX$.

Problem

Let $X$ be a continuous random variable with the following PDF $$\nonumber f_X(x) = \left\{ \begin{array}{l l} ce^{-4x} & \quad x \geq 0 \\ 0 & \quad otherwise \end{array} \right.$$ where $c$ is a positive constant.

1. Find c.
2. Find the CDF of X, $F_X(x)$.
3. Find $P(2 < X < 5)$.
4. Find $EX$.

Problem

Let $X$ be a continuous random variable with PDF $$\nonumber f_X(x) = \left\{ \begin{array}{l l} x^{2} + \frac{2}{3} & \quad 0 \leq x \leq 1 \\ 0 & \quad otherwise \end{array} \right.$$

1. Find $E(X^{n})$, for $n=1,2,3,\cdots$.
2. Find the variance of $X$.

Problem

Let $X$ be a $uniform(0,1)$ random variable, and let $Y = e^{-X}$.

1. Find the CDF of $Y$.
2. Find the PDF of $Y$.
3. Find $EY$.

Problem

Let $X$ be a continuous random variable with PDF $$\nonumber f_X(x) = \left\{ \begin{array}{l l} \frac{5}{32}x^{4} & \quad 0 < x \leq 2 \\ 0 & \quad otherwise \end{array} \right.$$ and let $Y = X^{2}$.

1. Find the CDF of $Y$.
2. Find the PDF of $Y$.
3. Find $EY$.

Problem

Let $X \sim Exponential (\lambda)$, and $Y=a X$, where $a$ is a positive real number. Show that $$Y \sim Exponential\left(\frac{\lambda}{a}\right).$$

Problem

Let $X \sim Exponential (\lambda)$. Show that

1. $EX^n=\frac{n}{\lambda} EX^{n-1}$, for $n=1,2,3,\cdots$;
2. $EX^n=\frac{n!}{\lambda^{\large{n}}}$, for $n=1,2,3,\cdots$.

Problem

Let $X \sim N(3,9)$.

1. Find $P(X > 0)$.
2. Find $P(-3 < X < 8)$.
3. Find $P(X > 5|X > 3)$.

Problem

Let $X \sim N(3,9)$ and $Y = 5 - X$.

1. Find $P(X > 2)$.
2. Find $P(-1 < Y < 3)$.
3. Find $P(X > 4|Y < 2)$.

Problem
Let $X$ be a continuous random variable with PDF $$\nonumber f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}} \hspace{20pt} \textrm{for all }x \in \mathbb{R}.$$ and let $Y = \sqrt{|X|}$. Find $f_Y(y)$.

Problem

Let $X \sim Exponential(2)$ and $Y = 2 + 3X$.

1. Find $P(X > 2)$.
2. Find $EY$ and $Var(Y)$.
3. Find $P(X > 2|Y < 11)$.

Problem

The median of a continuous random variable $X$ can be defined as the unique real number m that satisfies $$P(X \geq m) = P(X < m) = \frac{1}{2}.$$ Find the median of the following random variables

1. $X \sim Uniform(a,b)$.
2. $Y \sim Exponential(\lambda)$.
3. $W \sim N(\mu,\sigma^2)$.

Problem

Let $X$ be a random variable with the following CDF $$\nonumber F_X(x) = \left\{ \begin{array}{l l} 0 & \quad \textrm{for}\: x < 0 \\ x & \quad \textrm{for}\: 0 \leq x < \frac{1}{4} \\ x + \frac{1}{2} & \quad \textrm{for}\: \frac{1}{4} \leq x < \frac{1}{2} \\ 1 & \quad \textrm{for}\: x \geq \frac{1}{2} \end{array} \right.$$

1. Plot $F_X(x)$ and explain why $X$ is a mixed random variable.
2. Find $P(X \leq \frac{1}{3})$.
3. Find $P(X \geq \frac{1}{4})$.
4. Write CDF of $X$ in the form of
5. $$\nonumber F_X(x) = C(x) + D(x),$$ where $C(x)$ is a continuous function and $D(x)$ is in the form of a staircase function, i.e., $$\nonumber D(x)=\sum_k a_ku(x - x_k).$$
6. Find $c(x) = \frac{d}{dx}C(x)$.
7. Find $EX$ using $EX = \int_{-\infty}^{\infty} xc(x)dx + \sum_k x_k a_k$

Problem

Let $X$ be a random variable with the following CDF $$\nonumber F_X(x) = \left\{ \begin{array}{l l} 0 & \quad \textrm{for}\: x < 0 \\ x & \quad \textrm{for}\: 0 \leq x < \frac{1}{4} \\ x + \frac{1}{2} & \quad \textrm{for}\: \frac{1}{4} \leq x < \frac{1}{2} \\ 1 & \quad \textrm{for}\: x \geq \frac{1}{2} \end{array} \right.$$

1. Find the generalized PDF of $X, f_X(x)$.
2. Find $EX$ using $f_X(x)$.
3. Find $Var(X)$ using $f_X(x)$.

Problem

Let $X$ be a mixed random variable with the following generalized PDF $$\nonumber f_X(x) = \frac{1}{3}\delta(x + 2) + \frac{1}{6}\delta(x - 1) + \frac{1}{2}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}.$$

1. Find $P(X = 1)$ and $P(X = -2)$.
2. Find $P(X \geq 1)$.
3. Find $P(X = 1|X \geq 1)$.
4. Find $EX$ and $Var(X)$.

Problem

A company makes a certain device. We are interested in the lifetime of the device. It is estimated that around 2% of the devices are defective from the start so they have a lifetime of 0 years. If a device is not defective, then the lifetime of the device is exponentially distributed with a parameter $\lambda = 2$ years. Let X be the lifetime of a randomly chosen device.

1. Find the generalized PDF of $X$.
2. Find $P(X \geq 1)$.
3. Find $P(X > 2|X \geq 1)$.
4. Find $EX$ and $Var(X)$.

Problem

A continuous random variable is said to have a $Laplace(\mu,b)$ distribution [14] if its PDF is given by $$\nonumber f_X(x) = \frac{1}{2b} \exp \left( -\frac{|x - \mu|}{b}\right)\\ \nonumber = \left\{\begin{array}{l l} \frac{1}{2b} \exp \left( \frac{x-\mu}{b} \right) & \quad \textrm{if }\: x < \mu\\ \frac{1}{2b} \exp \left( -\frac{x-\mu}{b} \right) & \quad \textrm{if }\: x \geq \mu \end{array}\right.$$ where $\mu \in \mathbb{R}$ and $b > 0$.

1. If $X \sim Laplace(0,1)$, find $EX$ and $Var(X)$.
2. If $X \sim Laplace(0,1)$ and $Y = bX + \mu$, show that $Y \sim Laplace(\mu,b)$.
3. Let $Y \sim Laplace(\mu,b)$, where $\mu \in \mathbb{R}$ and $b > 0$. Find $EY$ and $Var(Y)$.

Problem

Let $X \sim Laplace(0,b)$, i.e., $$\nonumber f_X(x) = \frac{1}{2b} \exp \left( -\frac{|x|}{b} \right),$$ where $b > 0$. Define $Y = |X|$. Show that $Y \sim Exponential \left( \frac{1}{b} \right)$.

Problem

A continuous random variable is said to have the standard Cauchy distribution if its PDF is given by $$\nonumber f_X(x) = \frac{1}{\pi(1 + x^2)}.$$ If $X$ has a standard Cauchy distribution, show that $EX$ is not well-defined. Also, show $EX^2 = \infty$.

Problem

A continuous random variable is said to have a Rayleigh distribution with parameter $\sigma$ if its PDF is given by $$\nonumber f_X(x) = \frac{x}{\sigma^2} e^{-x^2/2\sigma^2} u(x)\\ \nonumber = \left\{\begin{array}{l l} \frac{x}{\sigma^2} e^{-x^2/2\sigma^2} & \quad \textrm{if}\: x \geq 0\\ 0 & \quad \textrm{if } x < 0 \end{array}\right.$$ where $\sigma > 0$.

1. If $X \sim Rayleigh(\sigma)$, find $EX$.
2. If $X \sim Rayleigh(\sigma)$, find the CDF of $X, F_X(x)$.
3. If $X \sim Exponential(1)$ and $Y = \sqrt{2\sigma^2X}$, show that $Y \sim Rayleigh(\sigma)$.

Problem

A continuous random variable is said to have a $Pareto(x_m,\alpha)$ distribution [15] if its PDF is given by $$\nonumber f_X(x) = \left\{ \begin{array}{l l} \alpha \frac{x_m^{\alpha}}{x^{\alpha+1}} & \quad \textrm{for}\: x \geq x_m \\ 0 & \quad \textrm{for}\: x < x_m \end{array} \right.$$ where $x_m, \alpha > 0$. Let $X \sim Pareto(x_m,\alpha)$.

1. Find the CDF of $X$, $F_X(x)$.
2. Find $P(X > 3x_m|X > 2x_m)$.
3. If $\alpha > 2$, find $EX$ and $Var(X)$.

Problem

Let $Z \sim N(0,1)$. If we define $X = e^{\sigma Z+\mu}$, then we say that $X$ has a log-normal distribution with parameters $\mu$ and $\sigma$, and we write $X \sim LogNormal(\mu,\sigma)$.

1. If $X \sim LogNormal(\mu,\sigma)$, find the CDF of $X$ in terms of the $\Phi$ function.
2. Find $EX$ and $Var(X)$.

Problem

Let $X_1$, $X_2$, $\cdots$, $X_n$ be independent random variables with $X_i \sim Exponential(\lambda)$. Define $$\nonumber Y = X_1 + X_2 + \cdots + X_n.$$ As we will see later, $Y$ has a Gamma distribution with parameters $n$ and $\lambda$, i.e., $Y \sim Gamma(n,\lambda)$. Using this, show that if $Y \sim Gamma(n,\lambda)$, then $EY = \frac{n}{\lambda}$ and $Var(Y) = \frac{n}{\lambda^2}$.

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