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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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about PDF of normal distribution function

Definition: X is said to have a normal distribution if its PDF is given by $f_X(x)=\frac{1}{\sqrt{2 \pi \sigma}} e^{\frac{(-(x-\mu)^2}{2 \sigma^2}}$ How do people come up with this, and how am I supposed to remember it?
shine
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Chi square and gamma distribution

Def: X has chi-square distribution with r degree of freedom if it has a gamma distribution with $\theta=2$ and $\alpha =\frac{r}{2}$, i.e. $f(x)=\frac{1}{\Gamma(\frac{r}{2})} 2^{\frac{r}{2}} x^{\frac{r}{2}-1} e^{\frac{-x}{2}}$, x>0. This is…
shine
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Distribution of means

Please help with a problem of practical application (explained by an example): Suppose there are 10 objects and 30 values (or scores). Suppose also there are 1000 students randomly selected from a large population. Now each student is required to…
user71758
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How to prove $f_{X+b}(x) = f_{X}(x-b)$?

In the probability course on Youtube at 6:23, the professor said that: $$f_{X+b}(x) = f_{X}(x-b)$$ Why is this the case? Could someone offer a proof?
SayMyNameHeisenberg
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Let $X \sim N(\mu,\sigma^2)$, define $Y=e^X$. Find CDF for $Y$ and then PDF (using the CDF)

(a) Find CDF for $Y$. (b) Then use the CDF to show that $Y$ has the following PDF $$f(y)=\frac{1}{y \sqrt{2 \pi \sigma^{2}}} \exp \left(-\frac{(\log y-\mu)^{2}}{2 \sigma^{2}}\right), \quad y>0$$ My problem is that I'm not allowing to use change of…
Sorry
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Find the probability of the product of continuous random and an indicator variable

Given that $X$ and $Y$ are jointly standard normally distributed, find the $Prob(X\cdot 1_{\{Y\leq c\}} \leq z)$. Where $1_{\{Y\leq c\}}$ is an indicator variable. c is a constant. $$P(X\cdot 1_{\{Y\leq c\}} \leq z)=P(X\cdot 1\leq z,Y\leq c) +…
Kofi
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What is an Hypergeometric distribution where the last event must be a success?

I'm trying to find out the name of a distribution that is like negative binomial, only for finite population and without replacement. Or like Hypergeometric distribution where the last event has to be a success. That is: Let's say we have $N$ balls…
Tal Galili
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Densities of stable distributions are smooth?

Fix $1
uniquesolution
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Show $Z$ is zero-mean and unit-variance

If the random variable $X$ has the mean $\mu$ and the standard deviation $\sigma$, show that the random variable $Z$ whose values are related to those of $X$ by means of the equation $$Z = \frac{X−\mu}{\sigma}$$ has $\mathbb{E}[Z] = 0$ and…
Sparkles the unicorn
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What does this $\infty$-like symbol mean?

What does this symbol (the $\infty$-like symbol but without the right arc) mean? (Marked by red arrow below) What does it say about the prior distribution?
Tomas
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A random variable as a function of another

Suppose we have a random variable $X$ and its distribution given by $f_X(x)$ and $Y = X^2$. Say $X = 1/2$ then is $f_{X,Y}(1/2,y) = f_{X,Y}(1/2, 1/4)$? If not, then how does it work?
Ray
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A probability exercise

I have a problem understanding the first part of this question : In a certain town, at time $t = 0$ there are no bears. Brown bears and grizzly bears arrive as independent Poisson processes with respective rates $\beta$ and γ. (a) What is the…
yassinemab
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Determine the value(s) of k for which p is a probability mass function

Determine the value(s) of k for which p is a probability mass function. Note that n is a positive integer. $$p(x) = kx, x = 1,2,3,... ,n$$ According to the solution manual, $k=\frac{2}{n(n+1)}$, but I don't know how to arrive at this answer. I…
LadymadeStar
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How to find the Variance of a generalised function of random variables?

Suppose I have a function $f(x_1, \dots, x_N)$, where $x_i$'s are random variables. $x_i$'s have SD $\sigma_i$, and are all independent. There are two additional assumptions, that $f$ is approximately linear within the range $x^{'}_i \pm \sigma_i$;…
sbp
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Is this a familiar distribution?

I have the following distribution but I am unsure if it is a common distribution. Assume that $\alpha +\beta>0$. I have searched but nothing comes up. Any help would be appreciated! \begin{equation} p(x) =…
dsmalenb
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