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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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$z_1 = \sqrt{-2\ln(x_1)}\cos(2\pi x_2)$ and $z_2 = \sqrt{-2\ln(x_1)}\sin(2\pi x_2)$ are standard normal random variables

Context: http://mathworld.wolfram.com/Box-MullerTransformation.html $x_1$ and $x_2$ are uniformly and independently distributed between $0$ and $1$. The article says that it can be verified that $z_1$ and $z_2$ are independent standard normal random…
man_in_green_shirt
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Is the negative binomial a possible way of solving this problem?

I am unsure if the negative binomial is applicable for this problem? Imagine a soccer team playing in a tournment with 5 games. The probability that they win (1/2), draw (1/6) , lose (1/3). At the end of the tournment the coach will lose his job if…
Curios and Knowledgeable
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Finding condtional distribution.

$X_1,X_2,$. . . . . . $X_k$ are independent $poisson$ random variables. We need to show that the conditional distribution of $X_1$ given $X_1+X_2+X_3+....+X_k$ is a binomial distribution. I started off with finding the $M.G.F$ of $(X_1 | Y=y)$…
User9523
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concept Marginal Probability

In the Above explanation it is written in square bracket that Note that $x-({1\over 2})\mathrm dx$ and $x+({1\over 2})\mathrm dx$ are the values of X and in this interval f(x,y) may be treated as constant , I did not understand why and how f(x,y)…
Milan Amrut Joshi
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Find the distribution of $Y=\log(1-F(x))$ if $X$ is a r.v with absolutely c.d.f. $F(x)$

If $X$ is a random variable with absolutely continuous distribution function $F(x)$, then find the distribution of $Y= \log(1-F(x))$
Shama
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How many numbers will I guess correctly?

I play a guessing game. In this game, there are 100 equally-sized, folded-up cards randomly dispersed in a bag. The cards are labeled 1 through 100. I draw out the cards one by one without replacement and try to guess the number on the card every…
Minden Petrofsky
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Sum of density functions

Let $f_2(t)$ and $f_3(t)$ be density functions associated to distributions $F_2(t)$ and $F_3(t)$. I have the following formula, $$\frac{(f_2(t)+f_3(t))/2}{(1-(F_2(t)+F_3(t))/2)}=\lambda$$ where this equations holds $\forall t$ and $\lambda$ is…
carrotcake
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Random variable distribution

2 cards are drawn from a 52 card deck. The random variable X represents the number of aces drawn. The random variable distribution: X = 0 : P(0) = 0.849 X = 1 : P(1) = 0.145 X = 2 : P(2) = 0.005 Is this how you do it? Probability of the draws: 0…
user395040
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Finding the distribution of the rounding error random variable

Suppose that a non-negative random variable $X$ has a distribution function $F(x)$, and that $Y$ is the rounding error if $X$ is rounded off to the nearest integer below. Show that $Y$ has the distribution function $$\sum_{j=0}^\infty [F(j+y) -…
ElleryL
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Find the cdf of the third smallest among $ X_{1},... , X_{8} $

We have $ X_{1},... , X_{8} $. All independent exponential r.v. with mean $ 1 $. I know how to find the cdf of the smallest among them, but i didn't see how to find the third smallest and its expected value.
Hal03
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Canonical form for marginal liklihood integral: random variable from exponentially modified gaussian with power mean-variance relationship

For a group of random variables $X_{i:\,0\leq i < k}$, I am interested in finding the liklihood $f(x\in X_i)=\text{P}(x|\text{model}$) given that $X_i \sim \mathcal{N}(\mu_i+\Theta_i, e^{\beta_0}(\mu_i+\Theta_i)^{\beta_1})$, $\Theta \sim…
deasmhumnha
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Prove f(x) is a valid pdf, and if x is a random variable with pdf f(x), find E(X).

Lads, I need some insight here. Given our $f(x) = \sqrt{\frac2\pi}e^{-(x^2/2)}$ where x is non-negative, I have to prove that $f(x)$ is a valid pdf. My work done so far: I've tried to pull out the constant, $\sqrt{\frac2\pi}$, out of the integral,…
KevinG
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Probability distribution about consecutive

Mangoes that are marketed by a particular orchard have masses which are normally distributed with mean mass $20.5g$ and standard deviation $4g$. The mangoes are packed into packets of $10$ mangoes per packet. What is the probability that $4$…
Heng Yong Wei
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Independent Geometric Distribution Function

Let X and Y be independent and geometrically distributed with the same parameter p. Compute $E(X|X+Y=k)$ for all k=2,3,... I tried to calculate $$\sum_{i=z}^P P(X)P(Y_z-x)/P(Z)$$ and then want to calculate $P(X+Y=k)$ but i am getting stuck in the…
Raveesh
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Marginal probability densities

I am trying to work on the following problem, and can't seem to quite figure out how to work out the marginal probability density functions: Given $ f(x,y) = \begin{cases} k & 0 \leq x \leq 2\,,\quad 0 \leq y \leq 1\,,\quad 2y \leq x…
ConfusedMathStudent
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