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

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2
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0answers
10 views

Difficulty in understanding pattern recognition and machine learning (Bishop)

I started with Pattern recognition and machine learning by Bishop, but after completing the first chapter(which took lot of time) I feel I am facing lot of problem in understanding the mathematics ...
0
votes
1answer
5 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) ...
0
votes
1answer
10 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
23 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
votes
0answers
40 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
1answer
18 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
24 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
0answers
17 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
25 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 $X$, ...
1
vote
0answers
18 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
37 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
38 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
0answers
26 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
vote
1answer
19 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 ...
0
votes
0answers
8 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 ...
0
votes
0answers
8 views

Product of two wrapped normal pdfs

Let $f(x; \mu_f, \sigma_f, \gamma)$ and $g(x; \mu_g, \sigma_g, \gamma)$ be Wrapped Normal PDFs with the same period $\gamma$, \begin{align*} f(x; \mu_f, \sigma_f, \gamma) = \frac{1}{Z_f}s_f &= ...
1
vote
0answers
15 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. ...
0
votes
0answers
6 views

How to solve for the prior probability distribution in this integral equation?

I've obtained the Bayesian posterior probability for a problem and found it to be equal to $$ z = \frac{\int_{0}^{1} p^{h'} p^{h} (1 - p)^t\Pr(p)\,dp}{\int_{0}^{1}\hspace{1.35em}p^h (1 - p)^t ...
1
vote
0answers
18 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?     ...
0
votes
1answer
24 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
12 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
51 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
29 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 ...
0
votes
1answer
24 views

Given a uniform $X$ distribution how can I find distribution of $Y=\ln(x)$? [on hold]

If $X$ is distributed uniformly on $(0,1)$ what is the distribution of $Y$ if $Y=\ln(X)$?
0
votes
2answers
24 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
11 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
vote
0answers
47 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
vote
1answer
9 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)$\dfrac{(1+e^t)^n}{2^n}$ ...
0
votes
1answer
21 views

Joint density function of hospital room charges [on hold]

In a model for hospital room charges $X$ and hospital surgical charges $Y$ for a particular type of hospital admission, the region of probability is: $$0 \leq Y \leq 2X + 1 \leq 3$$ The joint ...
0
votes
0answers
16 views

How to find the sum of a Normal and Gamma R.V

The pdf of a standard Gamma distribution is $f(x) = \frac{x^{\gamma-1} \exp(-x)}{\Gamma(\gamma)}$. How do I find the pdf of $Z = X + Y$ where $Y$ is the Normal distribution? I tried using the ...
0
votes
4answers
47 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 ...
0
votes
0answers
13 views

How to calculate marginal probability distribution for random variables following a given order

Assume that $x_1,x_2,\cdots,x_n$ are n random variables such that $x_1<x_2\le x_3\le x_4<\cdots\le x_n$. In the paper, "Non central distribution of ith largest characteristic rots of three ...
0
votes
1answer
22 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 ...
0
votes
1answer
12 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 ...
-1
votes
1answer
33 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
26 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
39 views

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

Consider two random variables, $X$ and $Y$, with the following properties: $X|Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$. Does $X|Y>y$ follow a normal distribution as well? If so, what are its ...
0
votes
0answers
13 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
votes
0answers
21 views

Order statistics difficult problem

The $n+1$ random variables $X_i$ ($1\le i\le n+1$) are independent and identically distributed with cummulative distribution $F$. Let $Y_k$the order statistics of $X_1,...,X_n$ and let $Z_k$ the order ...
1
vote
1answer
19 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
21 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
vote
0answers
25 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
vote
1answer
18 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
16 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
27 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 ...
0
votes
1answer
25 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 ...
0
votes
0answers
23 views

A variable within an equation is normally distributed. How to find probability?

I have an equation: Y = a(10^(1.5*x))^(2/3) where a is a constant (0.97) and x is normally distributed, with mean 1.18 and standard deviation 0.746. What I want to do is find the probability ...
2
votes
0answers
16 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 ...
-1
votes
1answer
26 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 ...
1
vote
0answers
6 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} < ...