Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

learn more… | top users | synonyms

0
votes
0answers
21 views

We place uniformly at random n points in the unit interval [0, 1]. [on hold]

How to go about the question when it asks: Denote by random variable X the distance between 0 and the first random point on the left. What is the probability distribution function FX(x) and pdf?
0
votes
1answer
34 views

Determine correlation and independence when only the joint density is given?

The joint pdf of $X = (X_1,\ldots,X_n)$ is: $$f_{X}(x_1,\ldots,x_n)=\begin{cases} Ar^2,&0 \le r \le R\\[0.2cm] 0,& \text{ otherwise }\end{cases}$$ where $r = \sqrt{x_1^2 + \ldots + x_n^2}$ ...
0
votes
0answers
25 views

Express expected value with help generating function

I understand, how to express expected value with help generating function. For example, I have the following generating function: $D(z) = p K(z) + q M(z)$, where $p + q = 1$. How can I express ...
2
votes
1answer
29 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
2
votes
1answer
23 views

Determine the density of this problem

Let $X$ and $Y$ be independent random variables with a common density. You know this density has support only within the interval $[a, b]$ and that it is symmetric around $(a + b)/2$ (but you are not ...
0
votes
1answer
23 views

Prove this random variable has support in the first quadrant only

Let $f(t)$ be a density with mean $\mu$ and variance $\sigma^2$ with support on the positive half line $(t>0)$. Now show $$g(x,y) = \frac{f(x+y)}{x+y}$$ has support only in the first quadrant. ...
0
votes
2answers
39 views

PDF of Gamma R.V. [on hold]

I know that $X \sim \exp(λ)$, $Z\sim \exp(λ)$ and $Y\sim \exp(λ)$ for $λ>0$. I also know that all three: $X, Y$ and $Z$ are independent. How do I find a pdf for $X+Z+Y$?
2
votes
1answer
21 views

Sum of uniformly distributed random variables in a given range

I am trying to find the sum of n uniformly distributed i.i.d random variables in the range [0-W]. I am aware that if the variables are distributed in the interval (0,1) then their convolution is given ...
-2
votes
1answer
30 views

Beta density function

In this problem, I need to use Beta density function to solve the integration. $$ \int_{0}^{100}x^{2}\left(\,100 - x\,\right)^{2}\,{\rm d}x $$ After applying $\,{\rm Beta}\left(\, 3,3\,\right)$ I ...
0
votes
5answers
39 views

How to integrate using known distributions

I'm having trouble figuring this integration out using known distributions. I don't know which distribution to use to solve this problem. It looks like a gamma to me. $$\int_{0}^{\infty} ...
1
vote
1answer
46 views

Expected value of a Poisson variable conditioned on sum [duplicate]

Setting $$X_1 \overset{d}{\sim} \operatorname{Poisson}(\alpha_1)$$ $$X_2 \overset{d}{\sim} \operatorname{Poisson}(\alpha_2)$$ $$S = X_1 + X_2$$ Find $E[X_1 | S =n]$ My argument is that since $X_1 + ...
0
votes
1answer
56 views

Expected value of this deceptively simple variable

Setting: $X \overset{d}{\sim} \pmb{U}[-1,1]$ and $$\begin{align*}&Y = |X|\\[0.4cm]& Z = \begin{cases}\dfrac{X}{|X|}, & \text{ if } X \neq 0,\\[0.2cm] 0,&\text{ otherwise ...
0
votes
2answers
28 views

Defining median for discrete distribution

In probability theory, a median of a probability distribution is a number $M$ such that the CDF of this distribution $F_\xi(x)$ satisfies $F_\xi(M)=\frac{1}{2} \tag1$ This works for continuous ...
4
votes
1answer
61 views

notation (ab)use for random variables, distributions, pdfs/pmfs

This question is about notation for random variables (RVs), distributions and pdfs/pmfs and their common (ab)use as I recently got confused. Let $X,Y$ denote random variables. First, notations I ...
0
votes
1answer
46 views

question on uniformly distributed random variable

Let $X$ be a uniformly distributed random variable on the interval $[0,10]$ and zero elsewhere and let $Y$ be another uniformly distributed random variable on $[0, 20]$ and zero elsewhere. Assuming ...
-4
votes
0answers
29 views

Find expectation and variance [closed]

Let $X$ be a uniformly distributed random variable on the interval $0<x<10$ and zero elsewhere and let $Y$ be another uniformly distributed random variable on $0<y<20$ and zero ...
-1
votes
1answer
27 views

Find the probability density function of $Y = 4X_1 – X_2$ [on hold]

Let $X_1$ and $X_2$ be independent normal random variables with means $23$ and $4$ and variances $3$ and $1$, respectively. Find the probability density function of $Y = 4X_1 – X_2$. No clue about ...
1
vote
0answers
16 views

Decisions on the order of integration with double integrals (when Deriving PDF via CDF) (Bank Problem)

Consider the following problem: Gandalf, Saruman and Radagast go to a bank together. There are two open counters which Gandalf and Saruman immediately go to get their service. Radagast goes to the ...
0
votes
0answers
27 views

Match this urn problem to a distribution

An urn initially contains r red balls and b black balls. A holding area outside the urn initially contain no balls. Balls are randomly chosen from the urn and: the chosen ball and the balls in the ...
0
votes
1answer
39 views

Distribution of transformed random variables

We have that f is a density w.r.t the lebesgue measure $m$ for a probability measure on $\mathbb{R}$, that f is continuous and strictly positive. X and Y are to random variables s.t. the distribution ...
0
votes
1answer
65 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
4
votes
1answer
109 views
+100

scores of individuals and evaluation

Suppose we have a fixed (ordered) set of $2000$ integers $p_m$ drawn from a discrete uniform distribution on $\{1,2,...,100\}$ arranged in a terrain. Let this terrain be denoted $\mathcal{T} = ...
0
votes
0answers
32 views

Birthday paradox variance of pairs [closed]

I tried solving following question but I am very stuck. Twenty people in a room each have independently random birthdays among $365$ possibilities. Let $P$ be the number of pairs of people that share ...
1
vote
1answer
49 views

Explain the result of this urn problem?

Suppose n balls are distributed in m urns. The probability that the first r urns receive k balls is $$\frac{\binom{n}{k}r^k(m-r)^{n-k}}{m^n}$$ I am most confused about the $r^k$ part. I know there ...
2
votes
1answer
26 views

distribution of the difference of discrete uniform RVs

Let $P_1, P_2$ be independent discrete uniform random variables on $\{0,1,...,k\}$. Suppose we want to compute $$\mathbb{P}(P_1 > P_2).$$ Is the best approach to see $\mathbb{P}(P_1 > P_2) = ...
0
votes
0answers
20 views

Montmort's card matching problem: Distribution of the number of matching cards?

(Introduction to Probability, Blitzstein and Nwang) Recall de Montmort’s matching problem from Chapter 1: in a deck of n cards labeled 1 through n, a match occurs when the number on the card ...
-3
votes
3answers
54 views

Calculate expected value. [closed]

Can someone give a hint for v). I don't know how to evaluate this integral from 0 to infinity. Thank you!
0
votes
0answers
29 views

Using gamma distribution to find the average duration of breaks after 10 calls with exponential distribution

Worker works 8 hours a day. Time between $ 2$ calls has $\exp(4)$ distrubution (expecting $4$ calls per hour). Duration of calls is $0$ (he just registers them). After $10$ calls he goes to $15$ ...
0
votes
0answers
49 views

Definite integral involving Error function

Let us write $$\mathrm{erf}(x)=\frac{2}{\sqrt {\pi}}\int_0^x e^{-t^2}dt $$ for the usual Gauss error function. Given natural numbers $m,n,k$ I am interested in computing the integral ...
0
votes
1answer
21 views

Resultant mean and variance of gaussian distribution

X be random Gaussian variable with mean u1 and variance v1. u1 itself is a random variable which is also gaussian distributed with mean u2 and variance v2. Then the distribution of X will be ...
0
votes
2answers
24 views

sum of two random variables with geometric distribution

hello. i've got 3 random variables, $X$, $Y$ with $GEO$ ~ $(p)$ for both , and $X+Y = Z$. i need to calculate $P(X | Z=k)$. so i started with: $P(X|Z=k)=P(X|X+Y=k)=P(x=j|X+Y=k)= ...
-1
votes
0answers
33 views

probability and bernoulii random variable? [closed]

A database file has 6,000,000 (six million) records, which occupy disk storage at a density of 12 records per block. A weekly update modifies 6.5 percent of the file and we assume that the changes ...
-4
votes
0answers
39 views

Using Bayes' Theorem,compute the probabilities [closed]

A manufacturing process produces computer chips of which $6$ percent are defective. This percent is actually found using a thorough (and expensive test) on a small random sample of chips. The plant ...
5
votes
2answers
50 views

probablity that $\max(X,Y)> a \min(X,Y)$

Two independent random variable $X$ and $Y$ having probability density functions uniform in the interval [0,1]. when $a \geqslant 1$, the probability that $\max(X,Y)> a \min(X,Y)$ is? (in terms of ...
0
votes
1answer
13 views

Marginal of Dirichlet distribution is Beta (integral)

Just for the sake of simplicity, take $K=3$ then a random vector $(X_1,X_2,X_3)$ has a Dirichlet distribution, i.e. $(X_1,X_2,X_3)\sim Dirichlet(\alpha_1,\alpha_2,\alpha_3)$ if the density takes the ...
0
votes
0answers
20 views

comparing a discrete and continous distribution

I'm looking for a real life example, and possibly a dataset, where one needs to compare a discrete r.v. $X$ with a continuous r.v. $Y$, for example by computing $P(X<Y)$. A special case of ...
0
votes
1answer
31 views

Binomial distribution, when variable isn't x

I've been using the formula $$p(x,N)=\frac{N!}{(\frac{N+x}{2})!(\frac{N-x}{2})!} p^{1/2(N+x)} q^{1/2(N-x)}$$ to determine the probability for a dog who walks in a straight line and can either move ...
0
votes
1answer
28 views

Exponential(1) distribution of Normally distributed X and Y

Let $X_1,X_2,X_3,X_4,X_5$ be a random sample from the uniform pdf: $f(x)= 1$, $0<x<1$ zero otherwise. Show that $\ln X_i$ has Exponential($1$) distribution for $i=1,2,3,4,5$. Solution: Let ...
0
votes
1answer
16 views

Mean and Variance of Nornally distributed distribution

Given X and Y be jointly normally distributed with $\mu_x=20, \mu_Y=40,\sigma_x=3, \sigma_Y=2$ and $\rho=0.6$. Find the mean and the variance of U=X+Y. soln: $U~N(\mu=60,\sigma^2=13). Am I right?$
1
vote
2answers
44 views

Probability generating function for urn problem without replacement, not using hypergeometric distribution

UPDATE: Thanks to those who replied saying I have to calculate the probabilities explicitly. Could someone clarify if this is the form I should end up with: $G_X$($x$) = P(X=0) + P(X=1)($x$) + P(X=2) ...
0
votes
1answer
27 views

Transforming a normal distribution to a uniform one

I'm searching for a method that transforms a normal distribution into a normal distribution. I've looked everywhere, but I'm not sure if I just missed something completely obvious, if this actually is ...
0
votes
1answer
53 views

How does the CDF come from the PDF

Let $X_{n}$ be an sequence of random variables s.t. $f(x)=1$ if $x=2+\frac{1}{n}$, $f(x)=0$ otherwise. Then the CDF is $F_{n}(x)=0$ if $x<2+\frac{1}{n}$ and $F_{n}(x)=1$ otherwise. My question is ...
0
votes
0answers
7 views

Calculate probability distribution $p\left(\left.X_{1:T}\right|Z_{1:T},y_{1:T}\right)$ in linear- non-Gaussian state space model.

I have a linear, non-Gaussian state space model. Observation equation: $y_{t}=a+bX_{t}+cZ_{t}+\epsilon_{t}$ $\,\,\,\,$ $\epsilon_{t}\sim\mathcal{N}\left(0,\omega^{2}\right)$ Transition equations: ...
3
votes
3answers
48 views

Independence of two normally distributed random variables

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
2
votes
1answer
35 views

A question about different pairs that are formed from a set of 16 different poeple such that…

I got the following problem: Given a set of 16 different people. We partition the people into pairs of two. Each pair needs to accomplish a task. And the probability that a pair accomplishes ...
1
vote
1answer
15 views

Suppose a random variable X has mean 0 and moment generating function as follows, find values of a and b

$M_x(t)=a(1+e^{-2t}+e^{-t} +e^t+be^{2t}), -\infty<t<\infty$ Do I take the first derivative of this function? How do I solve for two variables given only one equation? And as a followup ...
0
votes
2answers
32 views

Product of a Continuous and Discrete Distribution

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
0
votes
3answers
34 views

When to use Binomial Distribution vs. Poisson Distribution?

A bike has probability of breaking down $p$, on any given day. In this case, to determine the number of times that a bike breaks down in a year, I have been told that it would be best modelled ...
0
votes
0answers
20 views

Finding the limiting distribution $n\min(X_1, \dots , X_n)$ with uniformly distributed $X_i$

Find the limiting distribution of $nY_n$ where $Y_n = \min(X_1, ..., X_n)$ and $X_1, ..., X_n\sim \operatorname{unif}(0,2)$ are uniformly distributed random variables. Here is what I did: ...
14
votes
2answers
2k views

There are 10 men, 10 women, and 10 rooms. Each person randomly goes into a room.

What is the expected number of rooms with at least one man and woman? Our prof. gave us the following solution however, I'm confused about the probability portion of the answer (especially the ...