# Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

1,233 questions
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### Probability of success on third and fourth trials?

I have a pretty basic probability question, but I'm just having difficulties remembering what distribution this is. A coin is tossed until a head appears two times in a row. Given that we are using a ...
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### Average number of terms required in a sum of exponential variables to reach a specific limit

I have a sum $Y=\sum_{i=1}^{\infty}(X_i-t)u(X_i-t)$ where all $X_i's$ are i.i.d exponentially distributed random variables with parameter $\lambda$ and $t$ is a constant. I want to know how many term ...
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### How to show a binomial random variable dominates another binomial random variable with a smaller success value?

Let $X\sim B(n,p_h)$ and $Y\sim B(n,p_\ell)$ be two random variables following a respective binomial distribution, where $p_h>p_\ell$. I want to show that $$P(X\ge\alpha)\ge P(Y\ge\alpha),$$ for ...
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### Is $v=\ (r_i)^{-1}\cdot z$, a uniformly random value of a field?

We consider a finite field $\mathbb{F}_q$ where $q=2p+1$ and $q$ and $p$ are prime numbers. Let $r_i$ be a value picked uniformly at random from the field such that $r_i>\frac{q}{2}$. Let $z$ be a ...
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### Joint cdf and pdf of the max and min of independent exponential RVs [duplicate]

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y})$ $Z = \max({X,Y})$ Find the ...
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### Mixed Probability Distribution - Expected Value

Consider for one car owner the insurance policy with the following clauses: Deductible: If the loss $X>d$, then the insurer pays only for loss above $d>0$. Coverage Limit: If the loss $X>l$, ...
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### Maximum Likelihood Estimator of parameters of multinomial distribution

Suppose that 50 measuring scales made by a machine are selected at random from the production of the machine and their lengths and widths are measured. It was found that 45 had both measurements ...
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### Sum of Bernoulli random variables with different success probabilities

Let $X_{i} \in \{0,1\}$ be Bernouli random variable with probability of success $p_{i}$, i.e., $P(X_{i}=1) = p_{i}$ and $P(X_{i}=0) = 1-p_{i}$ and let $Y=\sum_{i=1}^{n}X_{i}$ for $n>0$. Is it ...
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### Multiplication of a random variable with constant

Suppose $X$ is a random variable which follows standard normal distribution then how is $KX$ ($K$ is constant) defined. Why does it follow a normal distribution with mean $0$ and variance $K^2$. ...
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### How to calculate $E[(\int_0^t{W_sds})^n], n \geq 2$

Let $W_t$ be a standard one dimension Brownian Motion with $W_0=0$ and $X_t=\int_0^t{W_sds}$. With the help of ito formula, we could get $$E[(X_t)^2]=\frac{1}{3}t^3$$ $$E[(X_t)^3]=0$$ When I try to ...
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### Distribution of weighted sum of Bernoulli RVs

Let $x_1,...,x_m$ be drawn from independent Bernoulli distributions with parameters $p_1,...,p_m$. I'm interested in distribution of $t=\sum_i a_ix_i,~a_i\in \mathbb{R}$ $m$ is not large so I can ...
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### Distribution of a random real with i.i.d. Bernoulli(p) binary digits?

Let $X_1, X_2, X_3, \ldots$ be an infinite sequence of i.i.d. Bernoulli($p$) random variables, and define the random real number $X = (0.X_1X_2X_3\ldots)_2$. Question(s): What can be proved about ...
Consider the following problem. Roll a die many times, and stop when the total exceeds $M$, for some prescribed threshold $M$. Call this time $\tau$, and call the running score after $n$ rolls $X_n$. ...
Assume that I have $n$ variables that are each $X_i \sim \text{Beta}(\alpha, 1)$ distributed (with the same $\alpha$, i.i.d.). Is there anything known about the distribution of the sum $Y=\sum_i X_i$?...