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Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
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{\Large \textbf{STA 312 f2023 Quiz 1}}\\
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\noindent
Let the random variable $X$ have an exponential distribution (see formula sheet on reverse), and let $Y=a \, X$, where the constant $a>0$. Derive the probability density function of $Y$. Show your work. Do not forget to indicate where the density is non-zero.


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{\Large \textbf{STA 312f23 Formulas}}\\   % 
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%%%%%%%%%%%%%%%%%%%%%%%%%%%% Common Distributions %%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
\begin{tabular}{|l|l|c|c|} \hline
\textbf{Distribution} & \textbf{Density or probability mass function} & \textbf{Expected value} &  \textbf{Variance} \\ \hline
Bernoulli($\theta$)  & $p(x) = \theta^x(1-\theta)^{1-x}$ for $x=0,1$ & $\theta$  & $\theta(1-\theta)$\\ \hline
Binomial($n,\theta$) & $p(k) = \binom{n}{k}\theta^k(1-\theta)^{n-k}$ 
    for $k = 0, 1, \ldots, n$ & $n\theta$  & $n\theta(1-\theta)$ \\ \hline
Geometric($\theta$) & $p(k) = (1-\theta)^{k-1}\,\theta$ for $k = 1, 2, \ldots$  & $1/\theta$  & $(1-\theta)/\theta^2$ \\ \hline
Poisson($\lambda$) & $p(k) = \frac{e^{-\lambda}\, \lambda^k}{k!}$ for $k = 0, 1, \ldots $ & $\lambda$  & $\lambda$ \\ \hline
Exponential($\lambda$) & $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$ 
%   \hspace{5mm} $F(x) = 1-e^{-\lambda x}$ for $x \geq 0$
 & $1/\lambda$  & $1/\lambda^2$ \\ \hline
Gamma($\alpha,\lambda$) & 
$f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} e^{-\lambda x} \, x^{\alpha-1}$ for $x \geq 0$ & $\alpha/\lambda$  & $\alpha/\lambda^2$ \\ \hline
Normal($\mu,\sigma^2$) & $f(x) = \frac{1}{\sigma \sqrt{2\pi}}\exp - \left\{{\frac{(x-\mu)^2}{2\sigma^2}}\right\}$  & $\mu$  & $\sigma^2$ \\ \hline
Chi-squared($\nu$) & $f(x) = \frac{1}{2^\frac{\nu}{2}\Gamma(\frac{\nu}{2})} e^{-\frac{x}{2}} \, x^{\frac{\nu}{2}-1}$ for $x \geq 0$ & $\nu$  & $2\nu$ \\ \hline 
\end{tabular} \vspace{2mm}

\begin{tabular}{lcl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Moments %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$E(X) \stackrel{def}{=} \sum_x x \, p_x(x)$ or $\int_{-\infty}^\infty x \, f_x(x) \, dx$
& ~ & 
$E(g(X)) = \sum_x g(x) \, p_x(x)$ or $\int_{-\infty}^\infty g(x) \, f_x(x) \, dx$    \\
$Var(X) \stackrel{def}{=} E\left( (X-\mu)^2 \right)$ 
& ~ &  
$Var(X) = E(X^2)-[E(X)]^2$ \\
If $X \sim N(\mu,\sigma^2)$, then $\frac{X-\mu}{\sigma} \sim N(0,1)$.
& ~ & 
If $Z \sim N(0,1)$, then $Z^2 \sim \chi^2(1)$. \\
\end{tabular}



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