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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How can $e^X \thicksim \mathrm{logN}(\mu, \sigma^2)$ given $X \thicksim N(\mu, \sigma^2)$ when they have different support?

According to Wikipedia (page about lognormal distribution), if $X \thicksim N(\mu, \sigma^2)$ then $Y=e^X \thicksim \mathrm{logN}(\mu, \sigma^2)$. But the support of $\mathrm{logN}$ is just $(0,+\infty)$ So assuming $N(0,\cdot)$ I would have a fair…
user2740
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X is uniformly distributed on the interval [1,4] and Y = sqrt(X)

Suppose that X is uniformly distributed on the interval [1,4], and that Y = sqrt(X). Evaluate E(Y) and Var(Y). I know the formulas to get the expected value and variance of a uniform distribution. I'm just not sure how to get from X to Y. Can I get…
user87654321
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Obtaining a distribution of values from an equation

Given any equation and range, for example, $y = x^2 + x$ where $x$ is a value from $0$ to $1$ (inclusive) Is it possible to determine the distribution of values outputted by this function between a give range of values? I can create a program that…
user4752555
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Hypergeometric distribution exercise!

A store has $20$ guitars in stock but 3 are defective. Claire buys $5$ guitars from this lot. (a) Find the probability that Claire bought $2$ defective guitars. I use $N=20,n=5, k = 3,x=2$ where $N$ is the total sample space, $n$ is the number of…
Joz
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Using PMF and CDF to calculate probability

Given the following CDF what is $$P(T > 3)$$ and according to my answer key it's 1-1/2 = 1/2. Can someone explain to me why it is 1-F(3), and would subtracting F(3) be subtracting 4 as well? Normally I saw stuff like t = 1,2,3,4,5,6...., but I've…
Belphegor
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Does joint pdf uniquely describes the random variables?

If two pairs of random variables $(X,Y)$ and $(U,V)$ have the same joint pdf $f_{X,Y}(x,y)=f_{U,V}(x,y)$, can we conclude that $(X,Y)=(U,V)$?
broke
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Exponential distribution question!

Suppose that the time between calls from your best friend has an exponential distribution with a mean time of $3$ days. (a) If you just received a call from her, what is the probability that you will receive the next call within the next $2$…
Jose soriano
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Is my probability reasoning here correct?

Sheldon Ross theoretical exercice A jar contains $n$ chips. Suppose that a boy successively draws chips from the jar, each time replacing the one drawn before drawing another. The process continues until the boy draws a chip that he has drawn…
CrispyBacon
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Probability distribution elimination

I thought I understood the problem until I read the solution. "Since the distribution function for $X$ is continuous (and differentiable) for $x>1$, it follows that $P[X=x]=0$ for $x>1$." How can this be when it says that $P[X≤x]=1-e^{-x}$ for…
Elie Fraser
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Distribution of $1/(1+X^2)$ if $X$ is standard Cauchy

Let $X$ be a Cauchy random variable with parameter $1$ i.e. with density $\dfrac{1}{\pi(1+x^2)}$. What is the density function of $Z:=\dfrac{1}{1+X^2}$? My attempt: Say $\phi(x) = \frac{1}{1+x^2}$ so then $\phi^{-1}(z) = \sqrt{\frac{1}{z}-1}$ and…
Luskentyrian
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Inverse cdf function of monotone transformations of random variable

Suppose $X\sim\mathcal{F}_X$ and denote $F_X(x)$ the cdf and $F_X^{-1}(x)$ the quantile function of $\mathcal{F}_X$ evaluated at $x$. Now define: $Y=\exp(X)$ and denote $\mathcal{F}_Y$ the distribution of $Y$. Since $\exp$ is a monotone increasing…
user42397
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Joining heterogeneous, discrete probability mass functions

Suppose we have a collection of discrete probability mass functions with different ranges, all of which are from 0 to some positive integer. As a simple example, we might be rolling 3 6-sided dice, 1 10-sided die, and 2 20-sided dice. (We'll ignore…
Jeff
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Distribution of arcsin of a uniform random variable

Question: Find the law of $\arcsin(X)$ where $X\sim Unif[0,1]$ and where $X\sim Unif[-1,1]$ My attempt: We say $f_X(x)=Unif[0,1]$, and that $Y=\arcsin(X)$ We say $x=\phi^{-1}(y)=\sin(y)$ and have $\frac{d}{dy}\phi^{-1}(y)=\cos(y)$ Then by following…
Luskentyrian
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determining distribution composed of uniform distributions

Let $X,Y,Z$ be i.i.d. $U(0,1)$ distributed. How can I determine the distribution of $$ \frac{X}{X+Y+Z}?$$ I have no idea how to go about this problem. Obviously this expression also has values between $0$ and $1$ but is there a way of finding the…
flawr
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Product of CDF and CCDF (or survival function)

Suppose we have two independent random Gaussian-distributed variables X and Y. X and Y represent thresholds for activation and deactivation, respectively. I'm interested in ensemble averaging over many iterations of X and Y and getting that…
Jeremy
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