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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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Does the CMF of the degenerate distribution have to be right continuous?

The CMF of the degenerate distribution at 0 is defined on Wikipedia as below: $$ F(x)= \begin{cases} 0 & \space \text{for} \space x<0 \\ 1 & \space \text{for} \space x\ge 0 \end{cases} $$ My question is, why do we define $F(0)=1$? Is it allowed…
Gnut
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Length of a line segment

A line segment of length 1 is cut into two pieces at a completely random point. What is the probability that the longer piece is at least three times the length of the shorter piece? Attempt Let X be the length of the shorter piece and Y be the…
Cal
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Finding the cumulative distribution function given density function

I have this statement: Let $f$ the probability density function of the continuous random $X$ defined by: $ f(x)=\left\{ \begin{array}{lr} \frac{x}{8} & 0≤x ≤4\\ 0 & \text{otherwise} \end{array} \right. $ My current development…
ESCM
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Probability distribution of people

There is have a group of 50 people where 30 are men and 20 are women and they are being separated into two equal classes of 25 people, what is the probability that any of the two classes will have 15 men and 10 women? Could you please help me solve…
LetMeLearn
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Probability of random variable given probability of its modulus

Assume that $z$ is a random vector. Also, assume that $$z = A(\|y\|)$$ where A is whitening function and $\|\cdot\|$ is complex modulus (norm) such that if $$y_j = a_j + ib_j,$$ then $$\|y_j\|=\sqrt{a_j^2+b_j^2}$$ and $y$ is a complex vector…
Blade
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Finding the Mode of a Probability Distribution Function

If I have a pdf represented by the following function: $$f(x) = \dfrac{3(x+x^2)}{14} $$ from $0
user3753
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Finding the Median of Distribution Function

Let a distribution function be defined as the following: $$F(x) = \begin{cases} 1 - \dfrac{16}{x^2} & x\geq 4 \\ 0 & x<4 \end{cases}$$ I was hoping for some confirmation on the median I calculated. I am very new to this material and…
user3753
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Prove that the following probability density is symmetric about zero.

This problem is question number 25 in John A. Rice's Mathematical Statistics and Data Analysis (3rd ed.). Let $X$ have the density $f$, and let $Y=X$ with probability $\frac{1}{2}$ and $Y=-X$ with probability $\frac{1}{2}$. Show that the density…
user41916
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If not a log-normal distribution, what distribution describes this data?

I have some empirical data which appeared to be best fit to a log-normal, but upon making a histogram of log(x) with 50 bins I got this result which appears to have some skew: What is the proper distribution to use for data whose log looks like the…
AAC
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Probability of new random variables that depent in 2 other random independent variable

You have 2 independent random variables with pdfs $$f_X(x)=0.25[u(x)-u(x-4)] \quad \text{and} \quad f_Y(y)=e^{-y}u(y).$$ Define new random variable by $$Z= \begin{cases} Y, & X \le 2\\ X, & X >2\end{cases}$$ Calculate:…
Knowledge
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Computing the joint distribution of correlated random variables

This is probably a stupid question but I will have to ask it. If you had a set of N correlated random variables and knew the correlation matrix, can one compute the joint probability distribution of all variables? Does it make a difference if the…
Bogdan
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Prove that for a fair coin, lengths of series of zeros or ones have geometric distribution

As in title: you're flipping a coin indefinitely (let's say heads gives 1, tails gives 0). E.g.: ${0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1\dots}$ How would you try proving that lengths of series of zeros or ones are of geometric distribution, that is…
lkky7
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Why are power laws with alpha between 2 and 3 common in real world networks

Many papers[1, 2] related to power laws say that alpha values between 2 and 3 are common in real networks. I understand that this means the mean is finite and the variance is infinite, but I do not have an intuitive understanding as to why this is…
BadProgrammer
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What is the mean and variance of the averaged value over $n$ random variables with Rice distributions?

Assuming there are $n$ uncorrelated random variables (RVs) with Rice (Rician) distributions $R1~N(u_1,s_1) \ldots R_n~N(u_n,s_n)$, with non-zero mean and different variance, what is the mean and variance of $Z= \displaystyle \frac{R1+\ldots+Rn}{n}$?…
Min
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How to understand particle size measurement and use them as input for simulation?

I am trying to understand the measurements of a particle size analyzer that could for example look like this: For this image I have three values for each error bar (min, max and mean). The y axis is here defined as $\frac{dN}{dlog(Dp)}$. Why is the…
Axel
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