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 to find the PDF of $Y$?

Consider the random variable $X$ which has CDF $F_X(x)$ and suppose further that the random variable $Y$ is a linear function of $X$, that is, $Y=aX+b$ where $a$ and $b$ are constants. We obtained the CDF $G(y)$ and $Y$ in terms of $F_X(x)$ when…
jahnavi punna
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How to find joint probability mass function of X and Y given marginal distribution of X and conditional distribution of Y given X.

how can we find joint probability mass function of discrete random variables X and Y, if we are given marginal distribution of X as well as the the conditional distributions of Y given X ? Thanks for the help.
Malick
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Given $y = sx, E[(x-E[x])(y-E[y])] = E[(x-E[x])(sx-E[sx])]$ how to derivate $Cov(x, y) = 0$?

Section 3.8 of "Ian Goodfellow and Yoshua Bengio and Aaron Courville. Deep Learning" says suppose we first sample a real number $x$ from a uniform distribution over the interval $[−1, 1]$. We next sample a random variable $s$. With probability…
JJJohn
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is there a detailed derivation of pdf of multivariate normal from the variance version to the covariance version?

wiki gives this equation to compute the pdf of the multivariate normal in k dimensions ${\displaystyle f_{\mathbf {X} }(x_{1},\ldots ,x_{k})={\frac {\exp \left(-{\frac {1}{2}}({\mathbf {x} }-{\boldsymbol {\mu }})^{\mathrm {T} }{\boldsymbol {\Sigma…
JJJohn
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Why is probability density funciton equal to 1 between 0 and 1?

I think I don't get the probability density function. At least on uniform distribution. There are infinitely many numbers between $0$ and $1$, in probability, I understand this means that the weight it assigns to individual points must necessarily…
Revolucion for Monica
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Could someone give an example to illustrate why it is greatest when n = m where draws number m from n numbered tickets?

this wiki page says Consider a case where n tickets numbered from 1 to n are placed in a box and one is selected at random; thus, the sample size is 1. If n is unknown, then the maximum likelihood estimator $\widehat {n}$ of n is the number m on…
JJJohn
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Could someone give an example to illustrate "a family of curves parallel to the x-axis"?

This wiki page says From a geometric standpoint, if we consider $f(x|\theta)$ as a function of two variables then the family of probability distributions can be viewed as a family of curves parallel to the x-axis, while the family of likelihood…
JJJohn
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What is the probability of getting 4 mistakes?

In a $12$-item multiple choice examination (each with four choices of which only one is correct), what is the probability of getting only $4$ mistakes assuming that no question is left unanswered? I used binomial distribution since each of the four…
PRD
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Evaluating mass of probability for a single point when the distribution is truncated

I have a problem with evaluating the probability mass at $K$ for truncated exponential disribution: $$ F(t)= \begin{cases} 0,& t<0\\ 1-e^{-\lambda t}, & 0\leq t
Novice
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Days required to catch fishes in a pond

A pond holds 4 fish. Each day a fisherman goes fishing and his probability of catching K = 0,1,2,3,4 fish that day follows a binomial PDF with p = 1/2.How many days should he plan on fishing so that the probability of his catching all 4 fish exceeds…
Taru Garg
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Linear replacement

Is it true that for any discrete random variable ξ the equality holds: $F_{aξ + b} (x) = F_{ξ}(\frac{x− b}{a})$ $∀x, a, b ∈ R (a \ne 0).$
Bully
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maximizing sum of area of rectangles

Let $(x_i)_{i=1}^{i=n}, (y_i)_{i=1}^{i=n}$ are sequence of positive numbers with $\sum_{i=1}^{n}x_i=\sum_{i=1}^{n}y_i=1$. Prove that $\sum_{i=1}^{n}x_iy_i$ is maximized when $x_i$ and $y_i$ are roughly the same size (More precisely, when…
Andy
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If $X$ follows Poisson distribution with mean $\lambda$ then what is the distribution of $X-2$?

If X is Poisson then I know the m.g.f is exp(λ[$e^t$−1]). And i need to find out the m.g.f of X-2. I am not able to reach the answer given. I am thinking X-2 is also Poisson with mean λ-2 but I am not sure whether I am right. Please help me.…
Azka
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Are $\delta(x - x^{(i)})$ and the indicator $\mathbf {1} _{X_{i}\leq t}$ the same?

Chapter 3 of the deeplearningbook gives this formula (equation_1) for "Empirical distribution" \begin{equation} \hat{p}(x) = \frac{1}{m} \sum_{i=1}^m \delta(x - x^{(i)}) \tag{3.28} \end{equation} Wiki gives this formula (equation_2) for "Empirical…
whnlp
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Writing a distribution as two uniform distributions

I have the following distribution: $$P(Z \leq z ) = \begin{cases} z/20 & 0 < x <10 \\ 1/2 + z/200 & 10 \leq z < 110 \\ 1 & z > 110 \end{cases} $$ How can I write this as a mix of two uniform distributions? For reference, the cdf of U(a,b) is (x -…
user702736
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