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

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0
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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
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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
50 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
36 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
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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: ...
15
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 ...
0
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0answers
18 views

Stationary Process Related To Balls in Urn

Suppose that you have an urn with 6 balls labelled '1' and 4 balls labeled '0'. At each time, you mix the balls and draw one, then record the value on the ball. You then replace it and add 2 more of ...
1
vote
1answer
25 views

Find Joint Distribution (marginal is known)

Got stuck...need help on the following question Given: $Y \sim f_Y(y)=\frac{1}{\theta }e^{-\frac{y}{\theta }}$, $y>0$. $R\sim g(r)$, $r>0$ $Y$ and $R$ are independent random variables. ...
0
votes
1answer
48 views

Mixed Distributions - Expectation and Variance

A bike has probability of breaking down $p$, on any given day. The repair cost of the bike, whenever it breaks down, is distributed as a Gamma random variable with shape $\alpha$ and rate ...
2
votes
1answer
15 views

Joint probability distribution of sum and product of two random variables

Let $X$ and $Y$ be two discrete random variables. I know the joint probability distribution of the vector $(X,Y)$, namely $P(X = x, Y = y)$ for all $x$ and $y$ in the sample spaces $\Omega_X$ and ...
1
vote
1answer
28 views

Central Limit Theorem approximation question

Suppose that the error, in grams, of a balance has the density $$f(x)=\frac{1}{4}e^\frac{-|x|}{2}$$ for $-∞<x<∞$, and that 100 items are weighed, independently of each other. Use the ...
0
votes
0answers
17 views

Product of two random variables - Resulting Distribution and Correlation?

Let $X \sim \mathcal{N}(0,1)$ and let $Z$ be a random variable independent of $X$ such that \begin{align*} P(Z=z) = \begin{cases}\frac{1}{2} & z=-1\\ \frac{1}{2} & z = 1\\ 0 & ...
1
vote
0answers
16 views

Is there a name for expressions that are invariant under the exchange of raw moments and cumulants?

I'm interested in expressions that are invariant under the exchange of raw moments and cumulants. This is trivially true of all expressions written only in terms of first order moments but nontrivial ...
0
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0answers
17 views

Trying to find similarity between collection of points

This is a kind of weird problem, and I'm not sure what the best Stack Exchange to post this on is, but I assume Mathematics could help the most. I have many sets of points in 3D space (xyz ...
0
votes
1answer
54 views

How to define the sum of two random variables

I wonder how to define sum of two random variables properly. Suppose $(\Omega,\mathcal{F},P)$ is a probability space and $X,Y\colon \Omega\to \mathbb{R}$ are random variables. It is most certainly ...
1
vote
1answer
16 views

Let $X$ and $Y$ be discrete random variables with joint PMF supported on the points $(0, 0), (1, 1),(1, 0), (1, −1).$

Let $X$ and $Y$ be discrete random variables with joint PMF supported on the points $(0, 0), (1, 1),(1, 0), (1, −1).$ (A) Assign positive probabilities to these four points so that $X$ and $Y$ are ...
0
votes
2answers
39 views

Finding the pdf of $(X+Y)^2/(X^2+Y^2)$ where $X$ and $Y$ are independent and normal

$X$ and $Y$ are iid standard normal random variables. What is the pdf of $(X+Y)^2/(X^2+Y^2)$? I am guessing you would transform into polar coordinates and go from there, but I am getting lost. ...
0
votes
1answer
18 views

Characteristic function of an unkown sum of random variables

$X_1,X_2,...\sim Pois(7), $ and independent random variables. $Y \sim Geom(1/4)$ independent from the $X_i$. My question is the characteristic function of: $X_1+X_2+...+X_Y$ Can someone tell me ...
0
votes
0answers
18 views

$(X,Y)$ is distributed uniformly over the square with vertices $(1,1)(1,-1)(-1,1)(-1,-1)$. Compute $\mathbb{P}(|X+Y|<1)$

$(X,Y)$ is distributed uniformly over the square with vertices $(1,1)(1,-1)(-1,1)(-1,-1)$. Compute $\mathbb{P}(|X+Y|<1)$ My ...
1
vote
2answers
46 views

Probability involving bread and jam!

SO, I drop a piece of bread and jam repeatedly. It lands either jam face-up or jam face-down and I know that jam side down probability is $P(Down)=p$ I continue to drop the bread until it falls jam ...
0
votes
0answers
9 views

Almost sure convergence of Chi-Squared variable

Setting $$X_1,X_2,\ldots \overset{d}{\sim} \mathcal{N}(0,1)$$ $$S = X_1^2 + \ldots + X_n^2$$ I would like to show $\frac{R_n}{\sqrt{n}} \rightarrow 1$ almost everywhere. I am using Borel-Cantelli ...
0
votes
0answers
17 views

Find Cumulative and the probability density function of Y

Usually I would integrate the function $y=x^2$ from 2 to 1 and to find the probability density function but I need to show it in terms of t. How do I do this? Also is the cumulative distribution = ...
1
vote
1answer
25 views

Distribution whose PDF is proportional to the product of a PDF and a CDF

Let $\phi$ be a symmetrical probability distribution function, that is nonnegative, that means $\int_{-\infty}^\infty \phi(x) dx = 1$ and $\phi(-x) = \phi(x) \forall x$ and $\phi(x) \geq 0 \forall ...
1
vote
1answer
31 views

Mixed Conditioning - Two Normal Distributions

Let $Z \sim \mathcal{N}(0,1)$ and $Y|Z \sim \mathcal{N}(Z, 1)$. Show that $f_{Z|Y}(z|y)$ is a normal density, and find the parameters of this density. What I have so far: \begin{align*} ...
1
vote
2answers
20 views

$X\sim U(0,1)$, $Y\sim U(0,2)$, how can I find CDF of $T=X+Y$ without knowing the joint PDF of $X$ and $Y$?

$X\sim U(0,1)$, $Y\sim U(0,2)$, how can I find CDF of $T=X+Y$ without knowing the joint PDF of $X$ and $Y$? Does anyone could help me with this?
-1
votes
1answer
26 views

$X,Y,Z$ are iid ~ $U(0,1)$, find $P(X>YZ)$ and $P(X<Y<Z)$

$X,Y,Z$ are iid ~ $U(0,1)$, find $P(X>YZ)$ and $P(X<Y<Z)$ I have no idea how to solve this problem, anyone could help me? Thanks
0
votes
0answers
15 views

Conditional expectations of joint normal distribution

$u_1$ and $u_2$ are jointly normal, with zero means, unit variances, covariance $\sigma _{12}$. I know $E(u_1|u_2)=\sigma _{12}u_2$, but why $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?
2
votes
1answer
37 views

Integral of a gaussian over a slice of the plane

I need to evaluate the following $n$ real integrals: $$\int_{\frac{\pi}{2}-\frac{\pi}{n}}^{\frac{\pi}{2}+\frac{\pi}{n}}\int_0^\infty\frac{1}{\pi \sigma^2}e^{-\frac{|re^{i\theta}-i|^2}{\sigma^2}} \ dr ...
0
votes
1answer
15 views

Independent binomial random variables with multiple parameters?

If $X$ and $Y$ are independent binomial random variables with parameters $n_1=3$ and $n_2=4$, and $p=0.3$(same for both), and $Z=X+Y$, what is $E(X^2 Y)$? I know that $E(Z)=(n_1 + n_2)p$ but I'm ...
0
votes
0answers
11 views

How to derive the Pdf for error function

Consider a stable causal, single-input/single output, linear time-invariant, discrete-time system. The noisy output is $y[n] = \sum_{i=0}^{p-1} c_i d[n-i] + w[n]$ where $c_i$ is the real-valued ...
0
votes
2answers
52 views

What's the expectation of square root of Chi-square variable?

Setting $$X_1,X_2,\ldots \overset{d}{\sim} \mathcal{N}(0,1)$$ $$V = X_1^2 + \ldots + X_n^2$$ Now I would like to find the expectation of $\sqrt{V}$. My biggest problem is I don't know how to write ...
0
votes
2answers
15 views

Probability Distribution Function for largest of $n$ uniformly random variables.

Generate $n$ random real numbers, from the uniform distribution on $[0,1]\in \Bbb{R}$. Let the largest such number be $X$. What is the probability distribution function of $X$?? I can plot ...
0
votes
0answers
11 views

Distribution of AYB in terms of distribution of Y

Let $A$ and $B$ be two random orthogonal matrices and let $Y$ be a random diagonal matrix. The distribution of $Y$ is known to be $p_y$. How can we express the probability distribution of $X = AYB$ in ...
2
votes
2answers
38 views

Why is this true? (sum of 2 uniform distributions)

If $X\sim U[0,1]$ and $Y\sim U[-1,0]$ and they are independent, then the distribution of $X+Y$ is not simply $U\sim [-1, 1]$, but it is the sum of 2 independent $U\sim [-0.5 ,0.5]$ distributions. Why ...
2
votes
1answer
20 views

Calculating PDF of $Z$ from $X,Y$ when $Z=X+Y$, given the PDFs of $X$ and $Y$

A Student is taking an exam which has two parts, X and Y, with each part given a score from 200 to 800. The students probability distribution for each part is given by $$ f_X(x)= \begin{cases} ...
0
votes
0answers
27 views

Probability distribution of the upper limit of an RNG given finite set of results

Suppose an unweighted random number generator outputs a random integer in the range $[0,u]$, and suppose I generate $n$ numbers and store the maximum number generated, $m$. Given only $m$ and $n$, can ...
0
votes
1answer
21 views

Help with integration of joint pdf to find $P(T_3 - T_2 \gt T_1)$

I have a joint pdf $f(x_1,x_2,x_3) = e^{-x_1 -x_2 -x_3}I(x_1 \ge 0)I(x_2 \ge 0)I(x_3 \ge 0)$ and I want to calculate the probability $P(T_3 - T_2 \gt T_1)$ where $T_1 \lt T_2 \lt T_3$ are the order ...
0
votes
1answer
17 views

Generalised probability density functions using Dirac deltas

I just cannot understand how a discrete random variable $X$ may be represented using a generalised probability density function (p.d.f.) $f_X$ as, \begin{equation}\tag{1} f_X(x) = \sum_{x_k \in R_X} ...
1
vote
0answers
76 views

distribution about the quotient of bunch of r.v.

Do you have idea about the distribution of: $$ Z=\frac{\xi_1}{\xi_1+\cdots+\xi_n} $$ where $\xi_i$ is i.i.d and standard Gaussian. Is there a technique to eliminate covariance and get the variance of ...
0
votes
2answers
23 views

Can a continuous random variable be jointly continuous with itself?

Furthermore, is it always the case? I don't see why not, but my issue is that $f_{XX}(x, x)$ looks weird. Does it accept two parameters, or only one?
0
votes
2answers
46 views

Finding an interval $I \subset \mathbb{R}^+$ such that $\phi$ is decreasing on $I$

Given $0<\alpha<\beta<1$, we define a function $$ \phi(x) = x - x \left[\frac{x^\alpha + x^\beta+1}{\alpha(x^\beta+1)+\beta(x^\alpha+1)} \right], $$ I am trying to find additional sufficient ...
2
votes
2answers
35 views

Formally proving that $E(|X|) < \infty \iff \sum_{n=1}^{\infty}P(|X|\geq n) < \infty$?

In other words, I have to prove that $\int_{-\infty}^{\infty}|x|f(x)dx < \infty \iff\sum_{n=1}^{\infty}P(|X|\geq n) < \infty$ where $f(x)$ is the density function. I know that the summation ...