Problems

01

PDF of sum from joint PDF

Suppose the joint PDF of $X$ and $Y$ is given by:

$$f_{X,Y}=\{\begin{array}{cc}{\displaystyle \frac{8}{81}}xy& 0\le y\le x\le 3\\ 0& \text{otherwise}\end{array}$$

Find the PDF of $X+Y$.

02

$★$ Poisson plus Bernoulli

Suppose that:

• $X\sim \mathrm{Pois}(\lambda )$
• $Y\sim \mathrm{Ber}(p)$
• $X$ and $Y$ are independent

Find a formula for the PMF of $X+Y$.

Apply your formula with $\lambda =2$ and $p=0.3$ to find $P_{X+Y}(7)$.

Solution

Solutions - 5220-02

(1) State the PMFs of $X$ and $Y$:

$$\begin{array}{ccc}{f}_X(x)=\{\begin{array}{ccc}\frac{e^{-\lambda }{\lambda }^k}{k!}& & k\in {\mathbb{Z}}_{\ge 0}\\ 0& & \text{else}\end{array}& & f_Y(y)=\{\begin{array}{ccc}p& & y=1\\ 1-p& & y=0\\ 0& & \text{else}\end{array}\end{array}$$

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(2) Derive the PMF of $S=X+Y$:

$$P[S=s]=P[X+Y=s]=P[X=s-y,Y=y]=P[X=s-y]P[Y=y]$$

Since $P[Y=y]>0$ only if $y=\mathrm{0,1}$, we have $P[S=s]=P[X=s]P[Y=0]+P[X=s-1]P[Y=1]$. Thus:

$$P[S=s]=\{\begin{array}{ccc}\frac{e^{-\lambda }{\lambda }^s}{s!}(1-p)+\frac{e^{-\lambda }{\lambda }^{s-1}}{(s-1)!}p& & s>0\\ e^{-\lambda }(1-p)& & s=0\\ 0& & \text{else}\end{array}$$

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(3) Evaluate at $\lambda =2$, $p=0.3$, $s=7$:

$$P_S(7)\approx 0.006015$$Link to original

03

PDF of sum of arbitrary uniforms

Suppose that:

• $X\sim \mathrm{Unif}[a,b]$
• $Y\sim \mathrm{Unif}[c,d]$
• $X$ and $Y$ are independent

Find the PDF of $X+Y$ in terms of the parameters $a,b,c,d$. You may assume that $b-a<d-c$.

04

$★$ Sums of normals

(a) Suppose $X,Y\sim 𝒩(\mu ,{\sigma }^2)$ are independent variables. Find the values of $\mu$ and $\sigma$ for which $X+X\sim X+Y$, or prove that none exist.

(b) Suppose $\mu =0$, $\sigma =1$ in part (a). Find $P[X>Y+2]$.

(c) Suppose $X\sim 𝒩(0,{\sigma }_X)$ and $Y\sim 𝒩(0,{\sigma }_Y)$. Find $P[X-3Y>0]$.

Solution

Solutions - 5220-04

(a)

(1) Suppose $X,Y\sim 𝒩(\mu ,{\sigma }^2)$ are IID. If $X+X\sim X+Y$, then $E[X+X]=E[X+Y]$ and $\text{Var}(X+X)=\text{Var}(X+Y)$.

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(2) Check the first condition:

$$E[X+X]=2\mu =E[X]+E[Y]=E[X+Y]$$

The first condition holds for any $\mu \in ℝ$.

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(3) Check the second condition:

$$\begin{array}{c}\text{Var}(2X)=4{\sigma }^2\text{Var}(X+Y)={\sigma }^2+{\sigma }^2=2{\sigma }^2\\ \\ 4{\sigma }^2=2{\sigma }^2\gg \gg \sigma =0\end{array}$$

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(4) Thus, $\mu \in ℝ$ and $\sigma =0$ (i.e., $X,Y$ are constants) are the only values that satisfy the condition.

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(b)

(1) Define $W=X-Y-2$:

$$\begin{array}{c}{\mu }_W=\mu -\mu -2=-2\\ \\ {\sigma }_W^2={\sigma }^2+{\sigma }^2=2\end{array}$$

Thus $W\sim 𝒩(-2,2)$, so $W=\sqrt{2}Z-2$.

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(2) Compute $P[X>Y+2]=P[W>0]$:

$$\begin{array}{c}P[W>0]\gg \gg P[\sqrt{2}Z-2>0]\gg \gg P[Z>\frac{2}{\sqrt{2}}]\\ \\ \gg \gg 1-\mathrm{\Phi }(1.4142)\gg \gg 1-0.9213\gg \gg 0.0787\end{array}$$

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(c)

(1) Let $W=X-3Y$:

$$\begin{array}{c}{\mu }_W=0\\ \\ {\sigma }_W^2={\sigma }_X^2+9{\sigma }_Y^2\end{array}$$

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(2) Compute $P[X-3Y>0]=P[W>0]$:

Since ${\mu }_W=0$, we have P[W > 0] = P[Z > 0] \quad \gg\gg \quad \colorbox{cyan}{ ParseError: Unexpected end of input in a macro argument, expected '}' at end of input: …colorbox{cyan}{0.5} ParseError: Expected 'EOF', got '}' at position 1: }̲

Link to original

05

PDF of sum of uniforms

Let $X$ and $Y$ be independent copies of a $\mathrm{Unif}[\mathrm{0,1}]$ random variable. Let $W=X+Y$.

Find the PDF of $W$.

06

Lights on

An electronic device is designed to switch house lights on and off at random times after it has been activated. Assume the device is designed in such a way that it will be switched on and off exactly once in a 1-hour period. Let $X$ denote the time at which the lights are turned on and Y the time at which they are turned off. Assume the joint density for $(X,Y)$ is given by:

$$f_{X,Y}(x,y)=8xy,0<x<y<1$$

Let $W=Y-X$, the time the lights remain on during the hour.

(a) Find the range of $W$.

(b) Compute a formula for the CDF of $W$, i.e. $F_W(w)$.

(c) Find the probability the lights remain on for at least 40 minutes in some given hour.