Let's say you have a sequence of i.i.d. random variables $r_1,r_2,r_3,\dots$ that are distributed as $\mathcal N(0,1)$. Now, consider the process where we sample a $k$ from $\text{Pois}(\lambda)$ and let $z$ equal $$\sum_{i=1}^{k}r_i$$  My question is what the distribution of $z$ is?  My first guess is that it would be $\mathcal N(0,\lambda)$. I do not how I would proof this, however. The variance might be slightly larger due to the uncertainty introduced by the Poisson distribution, for example. Proving the mean is $0$ is easy by symmetry.
120911
We have that $N$ and $X_1, X_2, \dots$ are all independent. We also have $\operatorname{E} [X_j] = \mu$ and $\operatorname{Var}[X_j] = σ^2$.  Then, we introduce an integer–valued random variable, $N$, which is the random sum such that: $$Z = \sum_{j=1}^{N+1}X_j.$$ Assuming that $N$ is distributed $\sim\mathrm{Poisson}(\lambda)$, what is the first moment and what is the variance of $Z$?  For a normal Poisson distribution, I know the variance is just $\lambda$, as is the mean. I'm having trouble understanding the implication of having the bounds be poisson distributed. Normally, I would just say "variance of the sum is the sum of the variance," but I don't think that's how it works with random sums. Any hints/guidance appreciated.
49180
Hy, I hope everyone is doing fine. Lately I have been studying the topic of probability, I am aiming to improve on it, recently i came across this hard problem.   Let $(u_{n})$ a sequence of random independent variables identically distributed following Rademacher distribution. And let  $f(x)=\sum_{n \geq 0} u_{n} x^{n}$.  Prove that : $$ f(x)~~\text{almost surely has infinitely many zeros  in}~~ [0,1].$$   I have been stuck on it for weeks now, but I did find out a hint to solve it on a forum by searching about it, Here is the hint:  Construct an increasing sequence $(x_{k})$ such that the event $A_{k}=\{f(x_{0}),...,f(x_{k}) \text{are not zero and have the same sign} \}$ are such that $p(A_{k+1})\leq \frac{6}{7} p(A_{k})$.      I find it difficult to construct by induction such sequence in a way that may permit us to solve the problem. Any proposition is welcome.
112321
Let $X_1, \ldots, X_n$ be i.i.d. exponentially distributed random variables with  density  $$\eqalign{\theta^{-1} e^{-x/\theta}, &amp;x \ge 0 \\ 0, &amp;x \lt 0} $$  and let $Y_i = X_{(i)}$ denote the order statistics such that $Y_1 \leq \cdots \leq Y_n$.  How to show that $$     2\frac{\left(\sum_{i=1}^{r}Y_i\right) + (n-r)Y_r}{\theta} $$     has a chi-square distribution with $2r$ degrees of freedom?  I wrote the joint density of $(Y_1,Y_2,...,Y_r)$ but nothing became apparent.
30668
The classic example... $X \sim Po\left (\lambda\right ), Y \sim Po\left (\mu\right)$, X and Y are independent.  Show that the conditional distribution of X is binomially distributed. Or in other words, $P(X=k\mid X+Y = n) = P (\tilde{X} = k), \tilde{X} \sim B\left (n ,\frac{\lambda}{\lambda + \mu}\right )$.   I've so far managed to reach to this step, and have been stuck since. Just somehow gotta get a $\frac{1}{n!}$ in the denominator, that would then complete the proof..or at least I think..  $$P(X=k\mid X+Y=n) = \frac{\frac{\lambda^{k}\mu^{n-k}}{k!(n-k)!}}{P(X+Y = n)}=  \frac{\frac{\lambda^{k}\mu^{n-k}}{k!(n-k)!}}{\sum_{i=1}^{n} \frac{\lambda^{i}\mu^{n-i}}{i!(n-i)!}}$$  Thanks for the help!
28011
Let $Z_i,Z_2,\ldots$ be IID Random Variables with density $f$. Suppose that $P(Z_i&gt;0)=1$ and that $\lambda=\lim_{x \to 0+} f(x)&gt;0$. How can I show that $X_n=n \times \min\{Z_i\}$ has a limiting exponential distribution with mean$1/\lambda$?  I know that the CDF of the first order Statistic is for $t&gt;0$  $$F_{min \{ Z_i \}} (t)= 1- \left[ 1-F(t) \right]^{n}$$  Convergence in distribution requires that $F_n (t) \to F(t)$ for some distribution function $F$ but I cannot quite take the limit of the above, with $t$ replaced by $x/n$, as the distribution is not known. I guess I have to insert the second piece of information somewhere but I do not recognise where.  Could you please give me a hint or two?  Thank you.
122555
The random variables $X_1,X_2,...X_n,Y_1,Y_2,...Y_n $ are independent and $U(0,a)$-distributed. Determine the distribution of $$Z_n=n\cdot \log\left(\frac{\max\left(X_{(n)},Y_{(n)}\right)}{\min\left(X_{(n)},Y_{(n)}\right)}\right)$$ where $X_{(n)},Y_{(n)}$ denotes $n$th order variable.  This problem is giving me a headache. The answer is that $Z_n$ is $\text{Exp}(1)$-distributed. But i can't derive that conclusion. I'd like to show you my derivation, but its so long and contains so many integrals and fractions that it would take a while to write it here. But basically my approach was to derive the distribution of $M=\max(X_{(n)},Y_{(n)})$ and $L=\min(X_{(n)},Y_{(n)})$. I get the following cdf's: $F_M=(F_X\cdot F_Y)^n=(F_X)^{2n}$ and $F_L=1-(1-(F_X)^n)\cdot (1-(F_Y)^n)=1-(1-(F_X)^n)^2$ ,  correct? $F_X $ stands for the cdf of $X_i$  Then i define the random variable $T=\frac{M}{L}$ and compute its cdf to finally be able to derive the distribution of $Z_n=n\cdot \log(T)$. But thats where i think everything goes wrong. Any help appreciated, thanks.
119800
I noticed that in the Normal distribution, the probability $P(x=c)$ equals zero, while for the Poisson distribution, it will not equal zero when $c$ is a non-negative integer.  My question is: Does the probability of any constant in the normal distribution equal zero because it represents the area under any curve? Or it is just only a rule to memorize?
100312
Let $X_1,X_2,...,X_n$ be i.i.d with df $F(y)=y^{\theta}, 0&lt;y&lt;1, \theta&gt;0$. Show that $\frac{X_{(i)}}{X_{(n)}}$, for $i=1,2,...,n-1$ and $X_{(n)}$ are independent.  I found the population pdf to be $$f(x)=\theta x^{\theta -1} I_{0&lt;x&lt;1}$$ Now, $$f(x_1,x_2,...x_n)=\theta^n\prod_{i=1}^{n}{x_{i}}^{\theta - 1}$$ Now, the joint distribution of the order statistics: $$f_{1,2,..,n}(x_1,...x_n)=n! \theta^n\prod_{i=1}^{n}{x_{i}}^{\theta - 1} I_{0&lt;x_1&lt;x_2,...&lt;x_n&lt;1}$$ Now , we take the transformation $Y_i=\frac{X_i}{X_n} $ for $i=1,2,..,n-1$ and $Y_n=X_n$ Have I done correctly so far? Also, I found out the Jacobian of transformation $={y_n}^{n-1}$. I am also finding it difficult to get the ranges of $Y_i$'s. Help!
88776
I have quite a simple question but I can't for the life of me figure it out.  For a set of iid samples $\,\,X_1, X_2, \ldots, X_n\,\,$ from distribution with mean $\,\mu$.  If you are given the sample variance as  $$ S^2 = \frac{1}{n-1}\sum\limits_{i=1}^n \left(X_i - \bar{X}\right)^2 $$  How can you write the following?  $$ S^2 = \frac{1}{n-1}\left[\sum\limits_{i=1}^n \left(X_i - \mu\right)^2 - n\left(\mu - \bar{X}\right)^2\right] $$  All texts that cover this just skip the details but I can't work it out myself. I get stuck after expanding like so  $$ S^2 = \frac{1}{n-1}\sum\limits_{i=1}^n \left(X_i^2 -2X_i\bar{X} + \bar{X}^2\right) $$  What am I missing?  Edit: A similarly equivalent expression is often given that I also can't derive but which may be more obvious is  $$  S^2 = \frac{1}{n-1}\left[\sum\limits_{i=1}^n X_i^2 - n\bar{X}^2\right] $$
119978
The likelihood function for the Poisson Distribution $P(X=k| \lambda)$, where $\lambda$ is the mean, has the property that the integral with respect to the mean integrates to unity (from $0$ to $\infty$): $$\int_0^\infty P(X=k|\lambda)\,d\lambda  = \int_0^\infty \frac{\lambda^kexp(-\lambda)}{k!}\,d\lambda = \frac{\Gamma(k+1)}{k!} = \frac{k!}{k!} =1.$$ This was shown and discussed in my .   Question: Does anyone know of any other probability density function (or probability mass function) which integrates or sums to unity in a similar way?  Thanks.
117265
This is an exercise of the Central Limit Theorem:     Let $Y^{\lambda}$ be a Poisson random variable with parameter $\lambda&gt;0$.   Prove that $\frac{Y^{\lambda}-\lambda}{\sqrt{\lambda}}\to Z\sim N(0,1)$ in distribution as $\lambda\to\infty$.   I've done that  $$ Z_n\to Z\sim N(0,1) $$ in distribution using the CLT, where $Z_n=(Y^n-n)/\sqrt{n}$. Some naive attempt to go is considering  $$ Y^{n}\leq Y^{\lambda}\leq Y^{n+1}\tag{*} $$ where $n\leq\lambda\leq n+1$ and somehow use the squeeze theorem. But both (*) and the squeeze theorem in convergence in distribution are NOT justified. How can I go on? Or do I need an alternative direction?
106599
If I have n samples from an $N(\mu,\sigma)$ distribution, how can I estimate the distribution from a subset of m of these n samples where the order within those n is known.  For example, if n = 100 and m is 3, I might know that one sample is 10 and 3rd highest of the 100, one is 5 and 42nd highest, and the last is 2 and 74th highest.
264
Let $X_1,X_2, \ldots , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0&lt;x&lt;a$. Let $X_{(1)},X_{(2)}, \ldots , X_{(n)}$ denote the order statistics. The range of the data is defined as $R = X_{(n)}−X_{(1)}$ and the midrange is defined as $V = \frac{1}{2}(X_{(1)}+X_{(n)})$ Derive the joint distribution of $(R, V )$ and deduce the marginal distributions of $R$ and $V.$  Attempt:   First I am attempting to generate the joint distribution of $(V,X_{(1)})$ and then integrate out the $X_{(1)}$.   For the joint distribution of $(V,X_{(1)})$ I have:   $$ f_{v,x_{(1)}}(v,x_{(1)})=2n(n-1)\int_{-\infty}^v [F(2v-x_{(1)})-F(x_{(1)})]^{n-2} f(2v-x_{(1)})f(x_{(1)}) \, dx_{(1)}$$  At this point I have no idea how to integrate out the $x_{(1)}$.  Edit: I have located the joint distribution of the range and mid-range:  $$ f_{r,v}(r,v) = n(n-1) \left[F\left(v+\frac{r}{2}\right) - \left(F\left(v-\frac{r}{2}\right)\right)\right]^{n-2} f\left(v-\frac r 2 \right) f(v+\frac{r}{2})$$  For each marginal distribution, I have to integrate out:  $$ f_r(r) = \int_{-\infty}^\infty n(n-1)[F(v+\frac{r}{2})-(F(v-\frac{r}{2})]^{n-2}f(v-\frac{r}{2})f(v+\frac{r}{2}) \, dv$$  $$ f_v(v) = \int_0^{\infty}n(n-1)[F(v+\frac{r}{2})-(F(v-\frac{r}{2})]^{n-2}f(v-\frac{r}{2})f(v+\frac{r}{2}) \, dr$$
18879
Let $U_1, \cdots, U_n$ be $n$ i.i.d continuous uniform random variables on $(0,1)$ and their order statistics be $U_{(1)}, \cdots, U_{(n)}$.  Define $D_i=U_{(i)}−U_{(i−1)}$ for $i=1, \cdots, n$ with $U_0=0$.  I am trying to figure out the distribution $\lim_{n\to\infty} \#\{i = 1, \cdots, n \mid nD_i \geq z\}/n$. This must be very well known, but I cannot find a reference.
85179
For two independent, discrete, uniformly distributed random variables $X$ and $Y$, I wish to obtain the distribution of the sum $Z=X+Y$. I have the densities: $$f_X(x)=\left\{\begin{matrix} \dfrac{1}{m+1},\;x\in [0,m]\\0,\;\text{otherwise}\end{matrix}\right.$$ $$f_Y(y)=\left\{\begin{matrix}\dfrac{1}{n+1},y\in [0,n]\\0,\;\text{otherwise}\end{matrix}\right.$$ $$f_Z(z)=\sum_{x=-\infty}^\infty f_X(x)f_Y(z-x)$$  I know that $f_Z(z)$ should look like a trapezoidal distribution. However, without knowing this, is it possible to calculate that distribution in a convenient way symbolically?   I'm finding it tedious to keep track of summation limits and such. For instance, separate cases for $m&gt;n$ and $m&lt;n$. I was wondering if there is a natural or intelligent notation for this, since this problem seems to be just so symmetric w.r.t $\,X$ and $Y$.  To make the question concrete: what is the most natural, symbolic/non-graphical way to solve for $f_Z(z)$? (I don't mind using another convolution formula if there is one, by the way)  Any help would be appreciated. Thanks.
110503
There is a sequence of independent random Poisson variables $Y_1, Y_2,\ldots$ with divergent parameter $Y_i \sim \text{Poisson}(\frac{1}{n})$. Does this sequence converge almost surely? In $L^2$ space? In probability?  I am really confused as I did some research on my own and almost every question I come up to has to do with the sum of a sequence; but I do not believe this has to do anything with the sum of $Y_n$ right? to prove it converges almost surely, I have to prove that $Y_n$ approaches the limit of $e^{-1}*\sum_{i=1}^{\infty} 1/i$? Any hints would be appreciated. Thanks.
13861
Suppose you have:  $X|N \sim \text{MN}(N,p_1,p_2...p_J)$  $N \sim \text{Poisson}(\lambda)$  Where is the marginal distribution of $X$?