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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.

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Find the distribution of $\ \min X_i $ where $\ X_i \sim Geo(p) $

Let $\ X_1, X_2, \dots X_n $ be independent variable with geometric distribution with parameter $\ p $ $\ (0 < p < 1 ) $ Find the distribution of $\ \min X_i , i = 1,2, \dots,n $ My attempt: $\ P\{\min X_i = j \} = P\{\min X_i \ge j \} - P\{\min…
bm1125
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Multivariate normal distribution clarification

It seems to me that if a random vector $X$ is to have a multivariate normal distribution, then it is necessary and sufficient that $X$ is a vector of independent Gaussians -- is this correct? In my understanding, a random vector $X$ of length $k$ is…
hessian
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Finding the Expectation and Variance, given the distribution function and density function for a continuous random variable

For this density function of a continuous random variable, X: $$f(x) = \begin{cases} c & \mbox{for } -1 ≤ x ≤ 1\\ \tfrac{c}{x^4} & \mbox{for } |x| > 1\end{cases}$$ and calculating its distribution function: $$F(x) = \begin{cases} \int_{1}^{\infty}…
Mike Campbell
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Finding the distribution function of a continuous random variable with a density function including a minimum

I have been scratching my head about this question for a long time. I found one other question on here that included a minimum function for probability functions but unfortunately there wasn't enough information to make me understand this enough to…
Mike Campbell
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calculate minimum sample number

i have assignment to solve the following problem, The height of a person is a random variable with variance ≤5 inches2. According to Mr. Chebyshev, how many people do we need to sample to ensure that the sample mean is at most 1 inch away from the…
Nour
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Chi-square with two degrees of freedom

How to derive chi-square distribution with $2$ degrees of freedom? $$X=z_1^2+z_2^2 \\ f(z_1,z_2)=\frac1{2\pi} \exp\left[\frac{-1}{2}(z_1^2+z_2^2)\right]$$ Consider the transformation $$z_1=r\cos(\theta), \qquad z_2=r\sin(\theta),$$ so…
Mee Al-Mee
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Why does marginal pdf give probabiltiy

Does the marginal pdf for X, $$\ f_X(x) = \int_{-\infty}^{\infty} f(x,y) \, dy$$ give us a probability if I put x in it? If so, I couldn't really understand. Because I thought that $\ f(x, y)$ does not give$\ P(X=x, Y=y)$. But here, I thought that…
whwjddnjs
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Weighted sums of power curve

I am not really sure how to even phrase this question, but here it goes: I'm looking at a distribution that follows a power curve (I think).. it looks something like this: f(0): 100000 f(1): 10000 f(10): 1000 f(100): 100 f(1000): …
math_idiot
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distributing k pennies to n children

there are 100 pennies and 10 children every child can get either 5, 10 or 20 pennies How many ways to do in this case? I assumed that n = pennies, and k = children so if first child can get 5, 10 or 20 pennies which is n-5, n-10, or n-20 if I keep…
ANDY
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Which software application can be used to find the PDF of the product of given two Random Variables?

I have a Random Variable $X$ whose PDF is not any of the standard distributions. Then, for the product of the random variable by itself say, $Z=X^2$, I can find the PDF manually. But, I have to solve for $20$ such PDFs which is becoming very tedious…
kunarapu priyatham
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Disaggregating Poisson

I have two random variables $x$ and $y$ that follow Poisson distribution with rate $\lambda_0$ and $\lambda_1$. Another random variable $z$ is defined as $(x+y)$ so it also follows a Poisson distribution with rate $\lambda_0+\lambda_1=\lambda_{01}$.…
Eln
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How can I find P[X ≥ 2Y] from the joint pdf?

Let X and Y have the joint pdf $f(x,y) =xye^{−x−y}$ $(x > 0,y > 0)$ Find $P[X ≥ 2Y]$. I really don't know what can I do for this...
Newt
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For which values of $p$ will the p.m.f. of the $\text{Bin}(n,p)$ distribution have its maximum at $n$?

My understanding is that this is equivalent to looking for values of $p$ such that the pmf is strictly increasing. The pmf of a binomial function is not easily differentiable though, so I doubt that's the right way to think about it. For $p$ close…
Hortense
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Use the MGF (Moment Generating Function) to find the joint distribution of $X$ and $Y$

Let $V$ and $W$ be independent standard normal random variables where $X=V+W$ and $Y=3W$ This is what I did: $$M_{x,y}(s,t)=E(e^{sx+ty})=E(e^{s(v+w)+t(3w)})=E(e^{sv+sw+3tw})=E(e^{vs+w(s+3t)})$$ and $$M_{v,w}(s,…
USC
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X and Y are Bernoulli. Suppose $P(X = 1, Y = 1) = P(X = 1) P(Y = 1)$. Prove that X and Y must be independent

Let X ∼ Bernoulli($\theta$) and Y ∼ Bernoulli($ψ$), where $0 < \theta < 1$ and $0 < ψ < 1 $. Suppose $$P(X = 1, Y = 1) = P(X = 1) P(Y = 1).$$ Prove that X and Y must be independent. Does it mean we have to prove $$P(X = 1, Y = 1) = P(X = 1) P(Y =…
Jeremy Yonatan
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