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

learn more… | top users | synonyms

1
vote
1answer
27 views

Distribution problem where |a|, |b|, |c|, and |d| are at most 10. Check my work?

How many ways can a+b+c+d=18, where a,b,c,d are integers such that $|a|,\ |b|,\ |c|,\ |d|$ are each at most 10? This is what I have so far. If all four numbers have the restriction -10 =< a, b, ...
0
votes
1answer
14 views

Does this integral $\int f_{X|Y}(x|y) dy$ has any meaning in probability or statistics

Suppose I have two random variables $(X,Y)$ with joint probability density given by $f_{X,Y}(x,y)$. Does integral \begin{align*} \int f_{X|Y}(x|y) dy \end{align*} evaluate to something or has ...
1
vote
1answer
31 views

Selecting n matches from two pockets.

Setting An eminent mathematician fuels a smoking habit by keeping matches in both trouser pockets. When impelled by need he reaches a hand into a randomly selected pocket and grubs about for a match. ...
0
votes
0answers
11 views

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
0
votes
1answer
14 views

Conditional distribution of geometric variables

Setting Suppose X1 and X2 are independent with the common geometric distribution w(k; p). Determine the conditional distribution of X1 given that X1 + X2 = n. Solution My argument is $$\Pr[X_1| ...
0
votes
0answers
28 views

Probability the pedestrian has to wait 3 time epochs to cross the street.

Setting A pedestrian can cross a street at epochs k = 0, 1, 2, . . . . The event that a car will be passing the crossing at any given epoch is described by a Bernoulli trial with success probability ...
5
votes
4answers
484 views

Probability that given a 1000 page book with 1000 misprints, a page will have 3 misprints.

Setting A book of 1000 pages contains 1000 misprints. Estimate the chances that a given page contains at least three misprints. Solution My solution is ...
0
votes
1answer
15 views

Ordering of elements drawn from uniform distribution

Setting $$X_1,\ldots,X_n \overset{iid}{\sim} \mathcal{U}[0,1]$$ Next order them so that $x_{(1)} \le x_{(2)} \ldots\le x_{(n)}$ Find $F_{(k)}(t) = \Pr[X_{(k)} \le t]$ in terms of a binomial sum, ...
1
vote
1answer
21 views

Assumptions of a probability distribution

Let $X$ be a continuous real-valued random variable indicating the fragility of a firm. Suppose that the firm defaults if $X$ takes a value above a threshold $u>0$. Hence $$ Prob(X>u) $$ is the ...
0
votes
0answers
9 views

Functions of random variables - bivariate case

this is the question: I approached the first question in this way: Then, for the second question: After, my friend told me that if Z is a Poisson distribution than Var(Z) should be 25. I ...
2
votes
1answer
25 views

Sufficient conditions for monotonicity with probability distributions

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ ...
0
votes
0answers
8 views

How write down PMF when random variable follows conditionally discrete uniform distributions with different support.

A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these ...
1
vote
1answer
22 views

Given the density function: $\frac{1}{2}\exp\left(-\frac{x}{2}\right), \space x > 0$ find $P\left(\sum_{i=1}^{81}X_i > 170\right)$

Suppose that $X_1,X_2...X_{81}$ are independent random variable with the same probability density function $$\frac{1}{2}\exp\left(-\frac{x}{2}\right), \space x > 0$$ Find ...
1
vote
2answers
23 views

Determine the expected value of a geometric distribution given some generic underlying distribution.

This is a variation of the standard waiting time problem. Suppose you have a sequence of variables $$X_0,X_1,X_2,\ldots \overset{iid}{\sim} F(x)$$ where $F(x)$ is continuous. And random variable ...
2
votes
1answer
40 views

Prove or disprove convergence in distribution of a poisson variable.

Let $$S \overset{d}{\sim} Poisson(\lambda).$$ I would like to determine $\frac{S-\lambda}{\sqrt{\lambda}}$ converges in distribution as $\lambda \rightarrow \infty.$ So my set up is: $$\Pr\left[a ...
0
votes
1answer
18 views

Given an unfilled pmf, How to compute the Correlation coefficient?

This is the format in which I was given the PMF. Sorry for the messy table, don't know how else to make a table. Given this pmf $x$$y$ $f_{xy}(x,y)$ 1       ...
1
vote
1answer
35 views

Probability of a point from one normal distribution being higher than a point from another independent normal distribution

Given two independent normal distributions: Distribution 1: Mean $= 23.95$, SD $= 7.44$ Distribution 2: Mean $= 16.29$, SD $= 7.79$ How often on average will a point from Distribution 2 be greater ...
1
vote
1answer
42 views

Check for independence of variables when the density (or distribution) is known.

This question is closely related to a previous one: Determine correlation and independence when only the joint density is given? Nonetheless, the setting is reproduced below. The joint pdf of $X = ...
2
votes
2answers
25 views

Density function and Integration to $1$

I have a function that's continuous and strictly positive on $\mathbb R$(it's also a density function w.r.t lebesgue to a probability measure), how do I go about defining it if I have the following ...
2
votes
1answer
32 views

At least 2 girls between every pair of boys distribution question?

Three boys and eight girls are seated randomly in a row of 11 chairs. All orders are equally probable. What is the probability that there are at least 2 girls between every pair of boys? What is ...
0
votes
0answers
17 views

How to Justify the exclusion of some samples?

I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and ...
0
votes
0answers
20 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
33 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
23 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
28 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
22 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
22 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
38 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
20 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
28 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
42 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
55 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
60 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 [on hold]

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
26 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
14 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
25 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
38 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
86 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 [on hold]

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
47 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
19 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
53 views

Calculate expected value. [on hold]

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
28 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 $15$ ...