Let $X, Y$ be independent r.v with moment generating functions $M_X(t)$ and $M_Y(t)$ respectively. Is there a function of $X$ and $Y, Z$, with moment generating function $$\frac{M_x(t) + M_y(t)}2$$

whats the basic difference between mutually exclusive events and independent events ? how do both associate with each other ? and can both of them occur simultaneously ?

Let $X_i$ be a $\{0,1\}$-valued r.v. with $P(X_i=1)=p_i$, $i=1,\dots, n$. $\{X_i, i=1, \dots, n\}$ are independent. Is there a name for the distribution of $\sum_{i=1}^n X_i$? How is $P(\sum_{i=1}^n X_i = m)$ determined? Some said $P(\sum_{i=1}^n X_i = m)$ is the coefficient of $x^m$ in $\prod_{i=1}^n (1-p_i + p_ix)$. I wonder why? Thanks!

Let $(X_n)$ and $(Y_n)$ be two sequence of random variables. $(X_n)$ and $(Y_n)$ are independent to each other. If $(X_n)$ and $(Y_n)$ have limits in distribution. $(X_n)$ tends to $X$, and $(Y_n)$ tends to $Y$ in distribution. Intuitively $X$ and $Y$ are independent, is this true? How can I prove it? Thank you very much.

Let’s say I have some experiment that I repeat $n$ times. Each time I repeat it, there is a $p$ chance that the outcome will be successful. If I set $X$ to be the amount of successful trials, how can I find the most probable and least probable $X$? I figured that the most probable […]

A few days ago in my Calculus BC class we were given a page of 6 challenging end of the year problems. That was a refreshing change from the drudgery we usually do (WebAssign). One of them went like this: There is a street of length 4 on which cars of length 1 wish to […]

Suppose we roll m dice, remove all the dice that come up 1, and roll the rest again. If we repeat this process, eventually all the dice will be eliminated. How many rolls, on average, will we make? The solution to this problem is $$ \sum_{n=1}^{\infty}n \cdot \left( \left(1-\left(\frac{5}{6}\right)^n \right)^m -\left(1-\left(\frac{5}{6}\right)^{n-1}\right)^m\right)$$ which can be rewritten […]

I’m working through some practice exercises for my probability course and I think there’s a concept I’m not totally understanding. The question I’m trying to work through is as follows: The number of mail a post office receives in one day is defined by a Poisson random variable with parameter $\lambda = 5$. The post […]

Let $ A $ and $ B $ be events such that $ P(A).P(B) > 0, $ then $ A $ and $ B $ are independent iff $ P(A|B) = P(A), $ or equivalently $ P(A \cap B) = P(A).P(B). $ $ \textbf{Question:} $ An urn contains $ m $ red balls and $ […]

For dices that we cannot distinguish we have learned in class, that the correct sample space is $\Omega _1 = \{ \{a,b\}|a,b\in \{1,\ldots,6\} \}$, whereas for dices that we can distinguish we have $\Omega _2 = \{ (a,b)|a,b\in \{1,\ldots,6\} \}$. Now here’s the apparent paradox: Suppose we have initially two identical dices. We want to […]

Intereting Posts

Solve $x^2$ $mod$ $23 = 7^2$
Asymptotic behaviour of a two-dimensional recurrence relation
If $r$ is a primitive root of odd prime $p$, prove that $\text{ind}_r (-1) = \frac{p-1}{2}$
Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?
Determine the convergence of $ \sum_{n=1}^{\infty}\left $
Is $\int_{-\infty}^{\infty} \sin x \, \mathrm{dx}$ divergent or convergent?
Can a polygon with minimal perimeter self-intersect?
Learning how to prove that a function can't proved total?
How to show that a measurable function on $R^d$ can be approximated by step functions?
Surjective map from polynomial ring over a field to the field.
Cover of (0,1) with no finite subcover & Open sets of compact function spaces
Finding the intervals where $f(x)=\frac{1}{|x-2|}-x$ is monotonous
eigen decomposition of an interesting matrix
General Formula for Equidistant Locus of Three Points
Why are generating functions useful?