Mathematics Stack Exchange
Mathematics Stack Exchange

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. Use this tag along with the tags (probability), (probability-theory) or (statistics).

Any probability distribution, including beta, binomial, chi, Erlang, gamma, geometric, lognormal, negative binomial, normal (Gaussian), Pareto, Poisson, Student's t, uniform, Wald, Weibull, zeta, and Zipf.

28080 questions
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Variance of jointly distributed random variables

Suppose $U$ and $V$ are jointly distributed continuous r.v's with $U \sim Uni(1,3)$ and $V$ given $U = u$ follows an exponential distribution with mean $\frac{1}{u}$. Calculate $Var(V \vert U)$. Attempt: Since $U \sim Uni(1,3)$, $f_U(u) =…
860009898987
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Setting up the integrand of a sum of iid random variables

In relation to this answer, why is the integrand $$f(x,z-x)=4x(1+x-z)$$ when trying to get the pdf of $Z=X+Y$ as $$f_Z(z)=\int 4x(1+x-z)dx$$ if $X$ and $Y$ are independent random variables where $X$ has a pdf given by $f_X(x)=2xI_{(0,1)}(x)$ and $Y$…
Antoni Parellada
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Integrating a joint distribution over only one variable -- a conditional moment?

What is the meaning of an integral like $$ \int dx_1 x_1 P(x_1,\dots,x_n)$$ for a joint probability distribution $P$? Can this be interpreted as the first moment of $x_1$ held conditional on the other $x_2, \dots, x_n$?
kevinkayaks
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Finding a marginal PDF

If we have a Random variable X that is $$2e^{-2x}$$ and Y that is an exponential distribution with X as its parameter $\lambda$, then how do we go around solving this question? I thought that we should find its CDF first by using $$P[Xe^{-XY} < Y]…
Logan X.
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Compute variance of a triangle distribution

How to compute the variance for the random variable Y that has the triangle distribution: $$f_Y(x) = \begin{cases} 4x,\;0\leq x\ \leq \frac12 \\ 2-4x, \; \frac12 \leq x \leq 1 \end{cases} \quad$$ The answer should…
LavO
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Let $Z_1,Z_2,Z_3$ be independent standard normal variables. Find their joint pdf and $P(Z_3>Z_1+Z_2)$.

Let $Z_1,Z_2,Z_3$ be independent standard normal variables. Find their joint pdf and $P(Z_3>Z_1+Z_2)$. If $X\sim N(\mu,\sigma^2)$, then $M_X(t)=e^{\mu t+\sigma^2t^2/2}$, $\infty
Karam
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Find the pdf of $Y=X^2$

Let $X$ have the uniform distribution $U(-1,3)$. Find the pdf of $Y=X^2$. As $X \sim U(-1,3)$, then the pdf of $X$ is $f:S_x \to \mathbb{R}$ defined as $f(x)=\frac{1}{4}$, where $S_x=[-1,3]$. Because $Y=X^2$, and because $-1\le x \le3$, we have…
Karam
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Grouping values in a bimodal distribution (beginner)

I'm confident that what I'm trying to do is textbook stuff, but I don't have the background to know what to look for, so I'm hoping someone will be kind enough to give me a reference so I can look up how to do this. I'm playing with digital signals…
Toby Eggitt
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Uniform Probability distribution question

Suppose $X$ is a continuous random variable with a cumulative distributive function of $F(x)$. Let $Y$ be a variable such that $Y$ can be represented in terms of $X$, i.e $Y=F(X)$. Show that the probability density function of $Y$ represents a…
Pen and Paper
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A binomial distribution problem

A factory manufactures watches and 100 watches are packed in a box. The probability that a watch is defective is given by 0.004. Find the probability that there are less than 3 defective watches in a box. answer: We need to find the probability…
LOLA
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What is the density of $1-X^3$ if $X$ is a Cauchy random variable?

What is the density function of $Y=1-X^3$, if $X$ is a Cauchy random variable? My approach: $$Pr(Y
Ryan
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Is this a Discrete uniform probability distribution?

This might be a simple question but I'm having trouble figuring out what kind of probability distribution this question is. Sam has a 2/3 chance of scoring a point each time she throws from the free-throw line in basketball. (You should assume that…
Bryan Hii
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Singular measures or finding density distribution

I'm a little confused with the next excercise: Let $F$ a probability distribution given by $$F(x)=\left\{\begin{array}{cc}0&\mbox{if}x<4\\1/5&\mbox{if }4\le x<6\\7/10&\mbox{if }6\le x<8\\23/30&\mbox{if } 8\le x<10\\1&\mbox{if }10\le…
sinbadh
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Find the joint pmf of X and Y. Prove that X and Y are not independent.

A fair coin is tossed three times. Let X denote a random variable that takes the value 0 if the first toss is a head and the value 1 if the first toss is a tail. Let Y be a random variable that defines the total number of heads in the three…
user976242
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How to decompose a gaussian into a sum of infinitely many gaussians?

Suppose that we have a normal distribution with known centre and variance. I would like to express it in terms of a sum of infinitely many gaussians whose centres are evenly, and infinitely densely, separated from minus to plus infinity. How can I…
artmyb
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