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

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0
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2answers
38 views

Find the distribution of $\sqrt{X^2+Y^2}$ where X and Y are normally distributed.

Find the distribution of $\sqrt{X^2+Y^2}$ where $X$ and $Y$ are independent normally distributed $\mathcal{N}(0,1)$. What is the best way to go about this? I tried finding the distribution of $X^2$ ...
1
vote
1answer
86 views

Summation of binomial/poisson question

A company has 800 customers, each with a probability 0.02 of dying during the next year (independent of all other policy holders). Find the probability that between 8 and 16 customers will die within ...
0
votes
1answer
245 views

Variance of a Joint Density Function

The random variables $X$ and $Y$ have joint density: $$ f_{X,Y}(x,y) = \begin{cases} 2-x-y,& 0<x,y<1\\ 0,&\text{otherwise}. \end{cases} $$ My question is to find $\operatorname{Var}(X)$...
0
votes
1answer
146 views

Find the distribution of the min(X,Y) where X and Y are independent and exponentially distributed.

Find the distribution of U=min(X,Y) where X and Y are independent random variables and both exponentially distributed with parameters lambda and mu respectively. The only headway I have made is that ...
2
votes
2answers
101 views

Moment Generating Function of a combination of 2 RVs

The number $N$ of cars sold has a poisson distribution with parameter $m$. Let $T=X_1 + X_2 +\cdots+ X_N$, where $X$ represents the size of the claim with a $\gamma(\alpha,\beta)$. $X_i$ is ...
0
votes
1answer
53 views

Deriving a distribution from the line generated from a point in a uniformly distributed circle and its origin

Let $a,r>0$ be two fixed numbers. A random point $(X,Y)$ is uniformly distributed over the circle {$(x,y) : x^2+(y-a)^2 = r^2$} with the centre of the circle at $(0,a)$. A line is drawn through $(X,...
0
votes
1answer
122 views

Poisson Distribution - dealing with big numbers

The mean number of defective products produced in a factory in one day is 21. What is the probability that in a given day there are exactly 12 defective products? I get some big numbers when I try ...
1
vote
1answer
17 views

To find the distribution of the random variable based on uniform distribution

Let $X_1,X_2,...,X_n$ be iid $U(-5,5)$ random variables. Then the distribution of the random variable $Y=-2\sum\limits_{i=1}^{10}\log(|X_i|/5)$ is (A) $\chi_{10}^2$ (B) $10\chi_{2}^2$ (C) $\chi_{20}^...
0
votes
1answer
62 views

joint distribution marginalization proof, is this right?

prove: $$p(x\mid z) = \sum_y p(x\mid y,z)p(y\mid z)$$ I understand a bit about marginalization. I think my prove should look like this: $$ p(x\mid z) = \sum_y p(x,y\mid z) = \sum_y p(x\mid y,z)p(y) $$...
1
vote
1answer
36 views

Distribution and density function of $Y=\frac{3X}{1-X}$

Let X be a random variable that is uniformly distributed on $[0,1]$. What are the distribution and probability density functions of $Y$ with $Y=\frac{3X}{1-X}$? I know that the density is the ...
4
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0answers
49 views

How to prove the following about eigenvalues

Let $\mathbf{M} = [m_{ij}]$ be a symmetric matrix of size $m\times m$ of real elements. Let $\mathbf{A} = [a_{ij}^R + ia_{ij}^I]$ be a random Hermitian matrix whose elements have variance, $\sigma^2$, ...
0
votes
2answers
95 views

exponential distribution with probability about texts

It is 9:00 p.m. The time until Joe receives his next text message has an exponential distribution with mean 5 minutes. A text has not arrived for 5 minutes. Find the probability that none will arrive ...
0
votes
0answers
80 views

Law of total Probability for conditional Probabilities

My question is, whether $P(AB|XY)=P(AB|XYZ)P(Z)+P(AB|XY\overline{Z})P(\overline{Z})$ is true, and why, and if it is true, is it true for any Partition? And can i apply ist to the continuous case by ...
-1
votes
1answer
77 views

Represent mode of a probability distribution.

I am finding it difficult to represent a simple analysis done in Matlab in equation form. The operation is of two step. Bin a data in 10 bins. (the distribution is unimodal) and then find the bin ...
0
votes
2answers
48 views

Marginal Distribution: Integrate a variable out

Suppose we have given the joint density $f_{(X,Y)}(x,y)$ of two random variables $X, Y$, where $f_{(X,Y)}(x,y)=g(x,y) \mathbb{1}_{y > t}$. Now we want to compute the marginal density of $X$, ...
3
votes
0answers
25 views

Can we write a Gaussian r.v. as $Y\mathrm{e}^{-\alpha U}$

I am facing the following problem. Let $\alpha>0$ and $U \sim \mathcal U ([0,1])$. Given a real valued random variable $Y$, independent of $U$ and admitting a density $f$ (wrt Lebesgue measure on $...
2
votes
2answers
204 views

Exponential random variable is almost surely finite

Let $T$ be a random variable with $Exp(\lambda)$ distribution for $\lambda >0$. I want to show that $T < \infty$ a.s. In order to do that, we need to show that $P(T < \infty) =1$. So I ...
1
vote
1answer
111 views

probability distribution , mean and covariance of balls in an urn

So I have the following question in "probability": An urn contains three balls: white, blue and red. At each stage a ball is picked up randomly and, if it is not red, it is returned to the urn. The ...
1
vote
1answer
98 views

Calculating the probability of something given the hazard rate function?

Suppose that the life distribution of a lightbulb of brand A has hazard rate function $λ_A(t) = t^{3}$ , t > 0. What is the probability that a brand A lightbulb burns out in less than 2 years?
1
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1answer
64 views

Clarifying the importance of the quantile function in probability theory

I want to cement my understanding of the quantile function in probability theory and here is the way I understand it. (1) We start off with some probability space $(\mathbb R, B = \sigma(\mathbb R),\...
0
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0answers
18 views

Which beta distribution(s) has a variance `V` and a skew `S`?

Let X be a beta distributed random variable with parameters $\alpha$ and $\beta$, variance V and skew ...
1
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0answers
43 views

Finding the conditional distribution of a normal RV given another normal RV

I'm having trouble with this question from a past qualifying exam: Question Suppose $Z \sim N(\mu,\sigma^{2})$, $W \sim N(0,1)$ and $V \sim N(0,1)$ are mutually independent normal random variables. ...
1
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0answers
35 views

Can a mixture of normals be a constant?

Q. Can a mixture of a finite number of 2-dimensional normal distributions, with different means and covariances, sum to a constant within some bounded region of the plane?       &...
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votes
1answer
100 views

Statistics normal distrution probability

A vending machine dispenses coffee into 8 ounce cups. The amount dispensed into these cups is normally distributed with a.mean of 7.6 oz and a standard deviation of 0.4 oz. a) Estimate the ...
0
votes
1answer
153 views

use poisson model to solve radioactive particles probability question

Suppose a radioactive source is metered for two hours, during which time the total number of alpha particles counted is 482. What is the probability that exactly three particles will be counted in the ...
0
votes
2answers
150 views

How do I compute the density of R?

A uniform random number X divides [0, 1] into two segments. Let R be the ratio of the smaller versus the larger segment. How do I compute the density of R?
0
votes
1answer
43 views

Positivity of pdf of sum of non-iid random variables

Suppose I have two random variables $X_i, i=1,2$ distributed on open subsets $U_i$ of a unit ball around $0$ in $\mathbb{R}^d$. Suppose $0\in U_i$ for every $i$. I assume that distribution of each $...
1
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2answers
138 views

Simple Explanation of Geometric distribution?

I really understood the explanation of Hypergeometric distribution by looking at this answer but when it comes to Geometric distribution I can't get how they calculate the probability distribution of ...
0
votes
1answer
14 views

Density Estimation and Analysis

This is an excerpt from BW Silverman's 'Density Estimation for Statistics and Data Analysis.' The oldest and most widely used density estimator is the histogram. Given an origin $x_0$ and a bin ...
1
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0answers
50 views

Showing the expectation of uniform r.v is $\frac{a + b}{2} $

Suppose $X$ is uniform over $(a,b)$, then $\mathbb{E} \{ X \} = \frac{a + b}{2}$. I am given that $$ \mathbb{E} \{ X \} = \int X P^X(dx) $$ where $P^X$ is the distribution of $X$. I am confused ...
1
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1answer
20 views

Calculating MGF for a random variable with pmf $P(X=x)=k\cdot( ^nC_{x})$

The pmf of a random variable X is given by $P(X=x)=k\cdot( ^nC_{x})$, $x=0,1,2,...,n$, where k is a constant. The moment generating function $M_X(t)$ is (A)$\dfrac{(1+e^t)^n}{2^n}$ (B)$\dfrac{2^n}{(...
0
votes
4answers
475 views

Conditional distribution of order statistics

Let $X_{(1)},...,X_{(n)}$ be the order statistics of a set of $n$ independent uniform $(0,1)$ random variables. Find the conditional distribution of $X_{(n)}$ given that $X_{(1)}=s_1,X_{(2)}=s_2....,...
1
vote
1answer
40 views

conditional probability has binomial distribution

Let $X_1, X_2$ be two independent random variables with $X_i \sim \mathrm{Pois}(\lambda)\,$ for $i=1,2$, where $\lambda>0$. Let $k,n \in \mathbb{N}$ and $0\leq k \leq n$. Define $f(k):=\mathbb{P}(...
0
votes
1answer
212 views

Exponential random variable with Minimum and Maximum Probability

If $X_1, X_2, X_3, X_4, X_5$ are independent and identically distributed exponential random variables with the parameter λ, compute (a) $P{(min(X_1,...,X_5) \le a});$ (b) $P{(max(X_1,...,X_5) \le a})...
-1
votes
1answer
293 views

Independent Sum Probability question

I have a question that I dont really know where to begin on any part. I have some ideas, but I am not sure about parts a-c. I think I should get d and e. Consider independent trials, each of which ...
0
votes
1answer
47 views

Expectation of a function of a normal random variable

Suppose $X\sim\mathcal{N}(0,1)$. I would like to find $\mathbb{E}[\frac{1}{\alpha+\beta X}|A<X<B]$ where $A, \alpha, \beta>0$. How should I go about it? Finally, if the answer is that there ...
1
vote
0answers
64 views

If $X\mid Y$ and $Y$ are both normal, is $X\mid Y>y$ normal as well?

Consider two random variables, $X$ and $Y$, with the following properties: $X\mid Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$. Does $X\mid Y>y$ follow a normal distribution as well? If so, what ...
0
votes
0answers
29 views

Probability distribution of $Z=X_1\,I(U<p) + X_2\, I(U\ge p)$?

Let $X_1$ and $X_2$ be two (possibly dependent) real random variables with distribution function $F_1$ and $F_2$, respectively. Let also $U$ be a random variable that is uniformly distributed over $[0,...
1
vote
1answer
41 views

Exponential order statistics

Let $X_1,...,X_n$ exponential random variables with parameter $\lambda$ and let $X_{(1)},...,X_{(n)}$ the order statistics of the random variables. I know that $X_{(1)}$ is exponential with parameter ...
1
vote
1answer
29 views

Let X be a non-negative continuous r.v. with pdf f(x)

Let $G(t)=\int_t^\infty$$f(x)dx$ Show that $E[X^2] = 2\int_0^\infty$$tG(t)dt$ I have not taken a course in probability in years and remember a theorem where X has a density function $f$ and ...
1
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0answers
63 views

Is it possible to derive a closed-form analytical expression when integrating over a triangular area of a bivariate Archimedean copula PDF?

Let's use, for example, one of the simpler Archimedean copulas - the Clayton copula with $\theta>0$. What I want to calculate is the probability associated with, say, a triangular region of ...
1
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1answer
32 views

Confused about binomial distributions

I'm confused about some simple binomial distribution problems in my textbook. Suppose p = 0.2 and n = 4. Calculate: P(x=2). The answer I got was 0.512 P(x<=2). The answer I got was 1.536 I am ...
1
vote
1answer
187 views

Definition and statistics of the Negative-Hypergeometric distribution

The Encyclopedia of Mathematics defines the Negative Hypergeometric distribution (NHG) in the following way: There are $N$ elements, of which $M$ are marked and the rest are unmarked. Elements are ...
0
votes
0answers
93 views

probability distribution of balls in an urn

So I have the following question in "probability": An urn contains three balls: white, blue and red. At each stage a ball is picked up randomly and, if it is not red, it is returned to the urn. The ...
1
vote
1answer
46 views

Modelling a continious-time queue which behaves differently when there are more or less people being served.

For my research I am trying to model a continuous-time queue which behaves differently when there are more or less people being served. The arrival rate in the queue is constant, however the departure ...
2
votes
0answers
38 views

what is the difference between joint probability distribution and random vector

Let $(S,\mathcal A, P)$ be a probability space and $\mathbf X:S\rightarrow \mathbb R^n$ random vector. Let $X_i:S\rightarrow \mathbb R$ be random variables such that $\mathbf X=(X_1,\ldots ,X_n)$. Is ...
0
votes
1answer
27 views

How to prove $P(A|B) = \sum_{i=1}{n} P(A|BH_i) P(H_i|B)$ if I know that $H_1, \ldots ,H_n$ is a complete system of events and $P(B)>0$.

How to prove $$P(A|B) = \sum_{i=1}^{n} P(A|BH_i) P(H_i|B)$$ if I know that $H_1, \ldots ,H_n$ is a complete system of events and $P(B)>0$. I know that when I have independent events $P(H_1,...,...
1
vote
0answers
14 views

Probability of summation of i.i.d. variables with a spherical joint distribution

I have a question regarding the probability of summed i.i.d. variables (log-returns) that have a joint spherical distribution. Obviously, the following statement holds: $$ P(X_1 + ... + X_{10} < -...
2
votes
1answer
293 views

How to solve this probabilty pizza delivery problem?

The company PizzaGo charges 6 dollars for one pizza. The delivery time $T$ is an exponential random variable with mean value $=m$ The company considers that a pizza's been delivered on time if $T <...
0
votes
1answer
49 views

Pdf of a linear transformation $f(x)=1-|x| ,-1<x<1$. Find pdf of $Y=X^2$. [closed]

Let $X$ be a continuous random variable with density function $$f(x)=\begin{cases} 1 - |x| &\text{if }-1<x<1,\\ 0 &\text{otherwise}\end{cases}.$$ Find the density function of $Y = X^2$. ...