Articles of probability

The sum of moment generating functions

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 $$\frac{M_x(t) + M_y(t)}2$$

Difference between mutually exclusive events and independent events?

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 ?

pmf of a generalization of binomial distribution

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!

Independence of two limits

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.

Most and least likely outcomes of a Bernoulli distribution experiment

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 […]

The Efficiency of Random Parking Problem

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 […]

dice problem – numerical approximation

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 […]

Probability Question: Poisson and pmf

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 […]

Are events $ A $ and $ B $ independent?

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 $ […]

Probabilistic paradox: Making a scratch in a dice changes the probability?

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 […]