2) Let \( X \sim \operatorname{Poisson}(\theta), Y \sim \operatorname{Poisson}(\lambda) \), independent. a. Find the distribution of \( X+Y \) b. Show that the distribution of \( Y \mid X+Y \) binomial with success probability \( \left(\frac{\lambda}{\lambda+\theta}\right) \) 3) Let \( X \) and \( Y \) be independent, standard normal random variables. Consider the transformation \[ U=X+Y \text { and } V=X-Y \] Show that the joint density of \( \mathrm{U} \) and \( \mathrm{V} \) is: \[ f_{U, V}(u, v)=\left(\frac{1}{\sqrt{2 \pi} \sqrt{2}} e^{-\frac{u^{2}}{4}}\right)\left(\frac{1}{\sqrt{2 \pi} \sqrt{2}} e^{-\frac{v^{2}}{4}}\right) \] 4) Let \( X \) and \( Y \) be independent, standard normal random variables. Consider the polar coordinates: \[ \tan \Theta=Y / X \text { and } R^{2}=X^{2}+Y^{2} \] Show that \( f_{R, \theta}(r, \theta)=\frac{1}{2 \pi} r e^{-r^{2} / 2} \) 1 5) Suppose a balance coin is tossed 3 times. Let \( X \) be the number of heads that we get in 3 tosses. Suppose another balance coin is tossed 2 times and let \( Y \) be the number of heads that we get in two tosses. Find the distribution of \( X+Y \) using the moment generating function method. 6) Consider independent normal variables \( \mathrm{Y}_{1}, \mathrm{Y}_{2}, \mathrm{Y}_{3} \) and \( \mathrm{Y}_{4} \) where \( \mu_{\mathrm{i}}=-3 \) and \( \sigma^{2}{ }_{\mathrm{i}}=2 \) for \( i=1,2,3,4 \). Let \( U=2 Y_{I}-5 Y_{2}+Y_{3}-2 Y_{4} \) a) Derive the MGF of \( U \) and identify the distribution of \( U \). b) Calculate \( P(U>9) \). 7) Consider independent exponential variables \( Y_{l}, \ldots, Y_{9} \) with \( \lambda=1 / 3 \). Drive the distribution of \( U=\sum_{i=1}^{9} Y_{i} \)
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2a) The sum of two independent Poisson random variables follows a Poisson distribution with parameter equal to the sum of the parameters of the two summed variables. Show more…
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Let $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample from a Poisson distribution with parameter $\theta>0$. From Remark 7.6.1, we know that $E\left[(-1)^{X_{1}}\right]=e^{-2 \theta}$. (a) Show that $E\left[(-1)^{X_{1}} \mid Y_{1}=y_{1}\right]=(1-2 / n)^{y_{1}}$, where $Y_{1}=X_{1}+X_{2}+\cdots+X_{n}$. Hint: First show that the conditional pdf of $X_{1}, X_{2}, \ldots, X_{n-1}$, given $Y_{1}=y_{1}$, is multinomial, and hence that of $X_{1}$, given $Y_{1}=y_{1}$, is $b\left(y_{1}, 1 / n\right)$. (b) Show that the mle of $e^{-2 \theta}$ is $e^{-2 \bar{X}}$. (c) Since $y_{1}=n \bar{x}$, show that $(1-2 / n)^{y_{1}}$ is approximately equal to $e^{-2 \pi}$ when $n$ is large.
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Functions of a Parameter
Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a Poisson distribution with parameter $\theta>0$ (a) Find the MVUE of $P(X \leq 1)=(1+\theta) e^{-\theta}$. Hint: $\quad$ Let $u\left(x_{1}\right)=1, x_{1} \leq 1$, zero elsewhere, and find $E\left[u\left(X_{1}\right) \mid Y=y\right]$, where $Y=\sum_{1}^{n} X_{i}$ (b) Express the MVUE as a function of the mle of $\theta$. (c) Determine the asymptotic distribution of the mle of $\theta$. (d) Obtain the mle of $P(X \leq 1)$. Then use Theorem $5.2 .9$ to determine its asymptotic distribution.
Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample with the common pdf $f(x)=$ $\theta^{-1} e^{-x / \theta}$, for $x>0$, zero elsewhere; that is, $f(x)$ is a $\Gamma(1, \theta)$ pdf. (a) Show that the statistic $\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}$ is a complete and sufficient statistic for $\theta$. (b) Determine the MVUE of $\theta$. (c) Determine the mle of $\theta$. (d) Often, though, this pdf is written as $f(x)=\tau e^{-\tau x}$, for $x>0$, zero elsewhere. Thus $\tau=1 / \theta$. Use Theorem $6.1 .2$ to determine the mle of $\tau$. (e) Show that the statistic $\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}$ is a complete and sufficient statistic for $\tau$. Show that $(n-1) /(n \bar{X})$ is the MVUE of $\tau=1 / \theta$. Hence, as usual, the reciprocal of the mle of $\theta$ is the mle of $1 / \theta$, but, in this situation, the reciprocal of the MVUE of $\theta$ is not the MVUE of $1 / \theta$. (f) Compute the variances of each of the unbiased estimators in parts (b) and (e).
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