Articles of probability theory

Why is $\mathbb{E}(\phi(X,Y) \mid X) = \mathbb{E}\phi(x,Y)|_{x=X}$ if $X$ and $Y$ are independent?

Let $X$ and $Y$ be two independent (real – valued) random variables , both defined on the probability space $(\Omega ,A,P)$ (a) $E|X| < \infty $ , $E|Y| < \infty $ . Let $g(.):R \to R$ (set of real number) be $g(x) = x + E[Y]$. Show that $E[X + Y \mid X] = g(X)$. […]

Mean of gamma distribution.

So I was trying to prove the mean result of gamma distribution which is $\frac{\alpha}{\lambda}$. My attempt, $E(X)=\int_{0}^{\infty }x f(x)dx$ $=\int_{0}^{\infty } \frac{\lambda^{\alpha}}{\Gamma (\alpha)}x^{\alpha}e^{-\lambda x}dx$ After integrating it, I got the result $$\frac{\lambda^{\alpha}}{\Gamma (\alpha)} \cdot\frac{\alpha}{\lambda}(\int_{0}^{\infty } x^{\alpha-1}e^{-\lambda x}dx)$$. I’m stuck here. Could anyone continue it for me and explain? Thanks a lot.

Sum of random variable

Considering two continuous random variables $X$ and $Y$ with $d.f \; F_X, F_Y$ I want to fin the distribution and distribution function of the sum $Z=X+Y$. \begin{align} P\{Z \leq z\} &= P\{X+Y \leq z\} \\ &=P\{X \leq z – Y\} \\ &=F_X(z – Y) \end{align} I know here that I have to use somehow the […]

It's a complicated case of simple dice roll.

Let T denote the number of times we have to roll a fair dice before each face has appeared at least once and let N denote the number of different faces appearing in the first six rolls. Then E(T|N=3) is?

Finding a Distribution When Introducing an Auxiliary Random Variable

Let (X,Y) be uniform on the unit ball; that is, $f_{(X,Y)}(x,y)=\begin{cases} \frac{1}{\pi}, &\text{if $x^{2}+y^{2}\leq 1$}\\ 0, &\text{otherwise.} \end{cases}$ Find the distribution of $R=\sqrt{X^{2}+Y^{2}}$. (Hint: introduce the auxiliary random variable $S=\arctan(\frac{Y}{X})$.) Any help you could give would be greatly appreciated!

Almost sure convergence

{Xn} is a sequence of independent random variables each with the same Sample Space {0,1} and Probability {1-1/$n^2$ ,1/$n^2$} Does this sequence converge with probability one (Almost Sure) to the constant 0? Essentially the {Xn} will look like this (its random,but just to show that the frequency of ones drops) 010010000100000000100000000000000001……

Central Limit Theorem for exponential distribution

Suppose that $X_1$ ….. $X_n$ are a random sample from a population having an exponential distribution with rate parameter $\lambda$. Use the Central Limit Theorem to show that, for large n, $\sqrt{n}(\lambda\bar{x}-1) \sim Normal(0,1)$ My attempt: honestly I am really not understanding what this question is asking. I can see that for an exponential distribution, […]

Expected value, I do not get this “wikipedia triviality”

on wikipedia are two definitions of the expected value for some random variable $E(X)=\int t f_X(t) dt$ and $E(X)=\int X dP$. I do not see how they are equivalent. In cases, that $X$ would be differentiable, then integration by substitution would do it. I mean honestly, wikipedia sounds as if this would be true for […]

Simplifying Multiple Integral for Compound Probability Density Function

Are there any ways to simplify this multiple integral? $$ \hat{f}\left(\left.y\right|\alpha\right)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\hat{f}\left(\left.y\right|\theta_{1}\right)\hat{h}_{1}\left(\left.\theta_{1}\right|\theta_{2}\right)\cdots\hat{h}_{K}\left(\left.\theta_{K}\right|\alpha\right)d\theta_{1}\cdots d\theta_{K} $$ Here, the density function $\hat{f}\left(\left.y\right|\theta_{1}\right)$ depends on parameter $\theta_{1}$ which is unknown and is governed by another density function, $\hat{h}_{1}\left(\left.\theta_{1}\right|\theta_{2}\right)$ with hyper-parameter $\theta_{2}$ which could again be governed by another density $\hat{h}_{2}\left(\left.\theta_{2}\right|\theta_{3}\right)$ with hyper-parameter $\theta_{3}$ and so no until we have density […]

Are there probability density functions with the following properties?

Given two distinct and continuous probability density functions on real numbers, $f_0$ and $f_1$ consider the following set of density functions: $$\mathscr{G}_0=\left\{g_0:\int_{\mathbb{R}}\log\left(\frac{g_0(y)}{f_0(y)}\right)g_0(y)\mathrm{d}y\leq \epsilon_0\right\} $$ and $$\mathscr{G}_1=\left\{g_1:\int_{\mathbb{R}}\log\left(\frac{g_1(y)}{f_1(y)}\right)g_1(y)\mathrm{d}y\leq \epsilon_1\right\} $$ for some sufficiently small $\epsilon_0$ and $\epsilon_1$ such that $\mathscr{G}_0$ and $\mathscr{G}_1$ are also distinct. In other words any density $g_0\in \mathscr{G}_0$ is not an element of […]