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

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1answer
14 views

Determine the density of sum of three normal variables.

Setting $\pmb{X} = (X_1,X_2,X_3)$ is a properly center normal with covariance matrix $$\begin{pmatrix} 3 & 4 & 0\\ 4 & 5 & 0\\ 0 & 0 & 6 \end{pmatrix}$$ Determine the ...
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1answer
22 views

distinguishing probability measure, function, distribution

I have a bit trouble distinguishing the following concepts: probability measure probability function (with special cases probability mass function and probability density function) probability ...
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1answer
22 views

Probability that there is sub-sequence of exact length

Can you help me to solve the following: Find probability that in sequence of N random uniformly distributed numbers there is increasing sub-sequence of exact length L.
3
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1answer
17 views

Probability measures and stochastically dependent events

If $P(B\mid A) > P(B)$ and $P(C\mid B) > P(C)$ can I infer that $P(C\mid A) > P(C)$? My suspicion is yes but I don't see how to prove it yet.
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3answers
16 views

probability of the empty set for arbitrary probability measures

I have a probability space $(\Omega, \mathcal{P}(\Omega), P)$. I want to know the probability of the empty set $\{\}$. Intuitively, I would say this probability is zero. It certainly is for the ...
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2answers
16 views

Product of two Beta distributed random variables

I have two Beta distributed random variables : $X_1=B(\alpha_1, \beta_1)$ $X_2=B(\alpha_2, \beta_2)$ What can we say about $Y=X_1.X_2$? Is this also a Beta distributed random variable?
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0answers
19 views

Convergence of $n^{-\gamma}T$ where $T$ a hitting time for uniform rvs, can I use CLT?

Let $X_1,X_2,\dots$ be iid uniform on $\{1,\dots,n\}$ and define $T=\inf\{k:X_k=X_r \text{ for some }r<k\}$. The objective is to figure out when $n^{-\gamma} T$ converges weakly to some ...
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0answers
17 views

Joint distribution of arrival times in Poisson process

I need to compute the following joint distribution in a Poisson process: $f_{S_A S_{A+B}}(t_1, t_2), t_2\ge t_1$ $S_A$ and $S_{A+B}$ are the arrival epochs of the $A^{th}$ and ${A+B}^{th}$ arrivals ...
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0answers
31 views

Binomial distribution giving me an answer above 1?

I am doing the following question. If i have a box of $20$ soccer balls and the independent chance of a soccer ball of being flat is $0.1$. What is the probability of having at least $4$ flat soccer ...
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0answers
7 views

Distribution of the ratio of two dependent chi-square

I look for my work the distribution of the ratio of two dependent chi-square variables $X, Y$ with different degrees of freedom for each one. Meanwhile I only found the distibution for the case where ...
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1answer
15 views

Finding the pdf of $X_1/(X_1+X_2)$ given $X_1,X_2 \sim \operatorname{Exp}(1)$

I have that $X_1,X_2 \sim \operatorname{Exp}(1)$. I need to find the pdf (probability density function) of $T$ where $T= X_1 + X_2$ and $R= X_1/(X_1+X_2)$. I convolved the pdf's of $X_1$ and $X_2$ to ...
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0answers
17 views

derivative of t distribution cdf wrt degrees of freedom

Given the cdf of a t distribution as follows: $T_\nu(x)=\frac{1}{2} + x\Gamma(\frac{\nu+1}{2}) + \frac{_2F_1 ...
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1answer
8 views

Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
0
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1answer
20 views

Determine the probability distribution of a ratio of two random variables?

Setting You are given two independent random variables $X_0,X_1$ with common exponential density $f(x) = \alpha e^{-\alpha x}$. Let $R = \frac{X_o}{X_1}$. Determine $\Pr[R > t]$ for $t > 0$. I ...
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2answers
42 views

The limit of an expected value vs expected value of a limit in this betting game

Setting The outcome $X$ of a slot machine takes values 1,2,or 3 with probability $p(1) = \frac{1}{2}$, $p(2) = \frac{1}{4}$, $p(3) = \frac{1}{4}$. We are given 3 for one odds, that is if we bet 1 ...
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0answers
13 views

Probability Density Function for Randomly Oriented Ellipse

I have an ellipse with a long aspect of a and a short aspect of b. The equation for this ellipse is found on this post: What is the general equation of the ellipse that is not in the origin and ...
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1answer
29 views

Central limit theorem in the setting of Poisson variables

Setting Given $S_{\lambda} \overset{d}{\sim} \operatorname{Poisson}(\lambda)$. Let $G_{\lambda}(t)$ be the distribution function of $\frac{S_{\lambda}}{\lambda}$. I need to determine ...
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0answers
17 views

Probability distribution for putting balls in boxes in a correlated way

I'm looking for help finding a probability distribution: Right now I have a problem where I have N indistinguishable balls, which I need to put into K indistinguishable boxes, each of which can hold ...
1
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1answer
39 views

Using the geometric distribution to find the probability that between 4 and 6 devices will be tested

Quality control tests spark plugs until they find one that doesn't work. If the probability of a spark plug working is 0.99, what is the probability that they will test between 4 and 6 (inclusive) ...
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1answer
31 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, ...
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2answers
22 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 ...
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1answer
32 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. ...
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0answers
19 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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1answer
22 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| ...
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0answers
29 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 ...
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4answers
1k 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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1answer
17 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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1answer
23 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 ...
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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 ...
2
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1answer
26 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 $$ ...
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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 ...
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1answer
23 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 ...
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2answers
25 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
45 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
36 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
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1answer
43 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
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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
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1answer
35 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
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0answers
18 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 ...
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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?
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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 $r = \sqrt{x_1^2 + \ldots + x_n^2}$ ...
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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
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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
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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
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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. ...
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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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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 ...