# Tagged Questions

Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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### A variation on the $F$-distribution

If I have $\frac{X/n_1}{Y/n_2}$ where $X$ and $Y$ are independent chi-squared random variables, with degrees of freedom $n_1$ and $n_2$, respectively, then the distribution of this ratio is given by ...
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### Result and proof on the conditional expectation of the product of two random variables

My problem is the following: $X$ and $Y$ are two random variables and $\mathcal{F}$ is a $\sigma$-algebra. Given that $X$ and $Y$ are independent, and that $X$ is independent of $\mathcal{F}$, can I ...
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### prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$

How to prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$. By memoryless property, $$P(X=i+s | X>i)=P(X=s)$$ How to get ...
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### How to calculate expected number of trials of this geometric distribution

I understand why the expected number of trials until there is a success is given by $$\sum_{i=0}^{\infty} i p q^{i-1} \ = \ E[\text{number of trials until} \ X=1] = \frac{1}{p}$$ where $p$ is ...
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### Uniform measure on the rationals between 0 and 1

I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the ...
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### PDF of product of variables?

could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? After having scanned ...
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### Evolution of a discrete distribution of probability

I am designing a virtual card game and I defined an evolution of probabilities, but I don't have the knowledge on this matter to find out how they will evolve. I hope you help me here, with ...
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### Random sample from discrete distribution. Find an unbiased estimator.

$X$ is a discrete random variable with parameter $a > 0$ whose pmf is defined as: $$f_X(x) = \begin{cases}0.2, &x = a\\0.3, &x = 6a\\0.5, &x = 10a\end{cases}$$ Say we have a random ...
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### How to prove these two random variables are independent?

If $X$ and $Y$ are independent Gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$ respectively, how to show that $U=X+Y$ and $V=X/(X+Y)$ are independent?
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### Normal approximation of tail probability in binomial distribution

From the Berry Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|\in O\left(\frac 1{\sqrt n}\right)$$ whereby $B_n$ has the standardized binomial distribution and $N$ has the ...
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### Probability distribution of tossing a coin until obtaining $k$ heads

My question is the following. We toss a coin, for which probability of obtaining heads is $p \in (0,1]$, until we obtain $k$ heads, not necessarily in a row (generally $k$ heads). Let $X$ be a number ...
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### If the sum of two i.i.d. random variables is normal, must the variables themselves be normal?

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random ...