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

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1answer
30 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
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1answer
13 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| ...
1
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1answer
23 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, ...
1
vote
1answer
99 views

Limiting distribution for $X_A$~ Beta distribution on [0, A] as A $\rightarrow \infty$ holding $E[X_A], Var[X_A]$ Constant

I am trying to determine the limiting form of a beta distribution as its range expands under isoparametric constraints on its first two moments.... For reference $X_A \sim Beta(0,A,\alpha,\beta) = ...
0
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1answer
13 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 ...
0
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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, ...
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4answers
318 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
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0answers
27 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 ...
6
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1answer
79 views

How Do I Find My Car

I have been discussing this problem with a coworker for a few days now and neither of us have made any headway on it. I would appreciate any help with a possible solution or maybe a suggestion of a ...
1
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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 ...
2
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1answer
24 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
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1answer
14 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
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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
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1answer
35 views

Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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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 ...
0
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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
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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 ...
2
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1answer
21 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 ...
1
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1answer
21 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 ...
2
votes
1answer
39 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
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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
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1answer
41 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 = ...
0
votes
1answer
32 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
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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 ...
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 ...
2
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2answers
24 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 ...
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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
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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 ...
4
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1answer
83 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} = ...
2
votes
1answer
27 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 ...
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
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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
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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 ...
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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 ...
1
vote
1answer
40 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 + ...
2
votes
1answer
326 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
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} ...
0
votes
1answer
54 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 ...
2
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1answer
41 views

Probability of Sample Variance Given Variance

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated! Here's the Problem: Suppose a considerable amount of ...
2
votes
1answer
47 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(X=0)$.Here is my solution: $$\mu= \frac {p(2) 2!}{p(1)}$$ Then how can find the mean? Thanks.
2
votes
3answers
38 views

Expectation of the function of a random variable

If a random variable $X$ has finite expectation, is the expectation of the function of $X$, e.g. $$f(X)=\exp(X)$$ also finite? How to prove or disprove?
2
votes
1answer
46 views

Binomial probability formula

The binomial probability formula goes like this, $$ f\left(x\mid p\right) = p^{x_i}\left(1-p\right)^{1-x_i} $$ But I wonder why the success probability and failure probability are multiplied? Can ...
14
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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 ...
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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 ...
0
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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 ...
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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 ...