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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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Joint moment generating function of (X,Y)

I'm looking for the E[XY] of the joint moment generating function of a couple of random variables (X,Y). We don't know if the variables are independent. I don't know how to solve the partial derivative ∂² E[e^(sX+tY)] ∂s ∂t
h1gh5ive
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Let Y be a random variable...

Let $Y$ be a random variable with density $$f_Y(y)=\frac32 y^2$$ if $-1< y <1$; zero otherwise. Find the density of $$U=Y+Y^2.$$ I have done up to $$f_U(u)=\frac{d}{du}F_U(u)=\frac{d}{du}P(Y+Y^2< u),$$ but am not sure how to go from there.
L.mak
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Find a new PDF by transforming a variable

I have a PDF $$ f_X(x) = \frac{1}{4}(x+1), \ \ \ \ \ \ \ \ \ 0 < x < 2 $$ I would like to use $$Z = \frac{9}{(X+1)^2}$$ and find a new PDF. Since, $z = g(x) = 9/(x+1)^2$ is monotone increasing, I thought I can use the following formula ($X$ has pdf…
user51966
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Gaussian from its moment generating function

By completing the square, a standard result is that the moment generating function (MGF) $\mathbb E e^{\beta X}$ of the standard Gaussian $X \sim N(0,1)$ is $e^{\beta^2/2}$. Is there a quick argument to show that if the MGF is $e^{\beta^2/2}$ then…
Martin Gale
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Can a probability density function be derived from any function?

I would like to confirm it is posible that a probability density function can be created ranging from $a$ to $b$. A PDF's area must be one within it's range so I just divide the function by it's area over that…
Garmekain
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how to calculate the probability of any statistic knowing X~N (statistical distributions)

In my documentation they explain me how to calculate specific probabilities of some statistics like $P(\bar X \ge x)$ or $P(S^2\ge x) $ knowing $X$~$N(\mu,\sigma)$ but then in the exercices they keep asking me things like this: $P(S^2/\sigma^2 \le…
Neku80
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How to verbally state a binomial distribution

When dealing with distributions (in Probability and Statistics), how do we verbally write a structure so that a screen reader would state it correctly for a visually impaired person? For examples with relevant parts bolded: "The standard normal…
Chelonian
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Why is this a Poisson problem?

Snowflakes are falling at an average rate of 10 flakes per square inch per minute. Calculate the probability that a 2 square-inch region has no snow flakes in a given 5 second time interval. How should I know to use Poisson distribution in this…
John Hoffman
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How to find a density function given two others

The question is Let U and V be two independent stochastic variables with $U\sim e(1)$ and with $P(V=1)=P(V=-1)=1/2$ Find the density function of W=UV First i want to find the distribution funktion as follows $F_W(w)=P(W\leq w)=P(V\cdot U \leq…
Emil Dare
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How to show that two normally distributed density functions are independent

Show that $U$ and $V$ are independent, where $U=X+Z$ and $V=X-Z$. I´m given that $X\sim N(0,1)$, $Z\sim N(0,1)$ and X and Z are independent. First I find U and V. This leads to $U\sim N(0,2)$ and $V\sim(0,2)$ From here I'm stuck, I know that…
Emil Dare
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Geometric distribution and the probability of getting success at "first try"?

If one knows the probability of success $p$, then how does one calculate the probability of getting success at "first try" using geometric distribution? Is it simply the probability of success? Are the successive events independent?
mavavilj
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How to write $\mathbb{P}(X_1 = s \mid X_2 = s )$ using c.d.f.s of $X_1$ and $X_2$?

Let $X_1, X_2 : \Omega \rightarrow \mathbb{Z}_{+}$ How to write $\mathbb{P}(X_1 = s \mid X_2 = s )$ using c.d.f.s of $X_1$ and $X_2$? I can get $\mathbb{P}(X_1 = s \mid X_2 = s)$ $= \frac{\mathbb{P}(X_1 = s \cap X_2 = s)}{\mathbb{P}(X_2 = s)}$ But…
mavavilj
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What is moment generating function of X1-X2

If X1, X2 are iid exp(1), Let Y=X1-X2, why moment generating function of Y is 1/(1-t^2)
user399659
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How to find limit on joint density

I am having trouble finding the integration limits here? suppose X and Y have joint density f(x,y) = c(x+y) for 0
Rohit Gulabwani
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Independent Random Variable?

The question is: Let $X$ and $Y$ which are independent Exp(1)-distributed random variables. Find the conditional distribution of $X$ given that $X+Y=c$ ($c$ is a positive constant). 1) I don't see how these can be independent if they are…
Squirtle
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