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

learn more… | top users | synonyms

0
votes
0answers
13 views

How to compute a probability expression (for a transition matrix of a Markov Decision Process)? (part 2)

I am creating a transition matrix (for a Markov Decision Process) and I am computing it using a Matlab script, which I am currently writing. My probability expression (for certain cases) looks like ...
0
votes
0answers
19 views

How to compute a probability expression (for a transition matrix of a Markov Decision Process)? (part 1)

I am quite new in the world of statistics, hence I am quite unsure when working with probabilities. I am creating a transition matrix (for a Markov Decision Process) and I am computing it using a ...
0
votes
0answers
9 views

Chernoff type bounds for negative binomial distribution

If I recall correctly I remember reading that we cannot get Chernoff type results for the negative binomial distribution because of something regarding lebesque measure. I don't quite know all the ...
1
vote
1answer
26 views

Expectancy of a joint density

A machine consists of two components, whose life times have the joint density function $ f(x,y)= \begin{cases} 1/50, & \text{for }x>0,y>0,x+y<10 \\ 0, & \text{otherwise} \end{cases} ...
0
votes
0answers
9 views

Decay time distribution with uniformly distributed source

Consider a kind of particle (source) that can decay into some other particle (product) with decay constant $\lambda$, i.e. the p.d.f is $f(t)=\lambda e^{-\lambda t}$, and the source is uniformly ...
1
vote
0answers
25 views

Survival probability (1D Brownian Particle)

Here is an interesting article from Wikipedia: First-hitting-time model I am particularly interested in how the following density is derived: $$p\left(x,t;x_0,x_c\right)=\frac{1}{\sqrt{4 \pi D ...
3
votes
2answers
30 views

Notation $E[t^X]$ where $X$ is a random variable

I have a quick question which occured in the context of probability-generating functions but maybe the issue is more basic. For a random variable $X$, the probability-generating function is given as ...
1
vote
2answers
25 views

Probability distribution of number of columns that has two even numbers in a chart

We distribute numbers $\{1,2,...,10\}$ in random to the following chart: Let $X$ be the number of columns that has two even numbers. What is the distribution of $X$? My attempt: ...
-2
votes
0answers
29 views

What is the right answer among the following four options? [closed]

probability calculus works upon joint occurance of the event under consideration alternative occurance of $d$ events under consideration both joint n alternative occurance of $d$ events under ...
0
votes
0answers
7 views

specific examples of random variables satisfying a given condition.

Theorems such as the central limit theorem only says random variables satisfying certain conditions have some properties. Now, what I am curious about is the existence of such random variables. For ...
-1
votes
1answer
28 views

Birth and Death process, CTMC, how is the solution here derived? [closed]

My question is about how the solution is reached, as I am completely lost on how. Any thoughts? Consider a birth and death process with birth rates $λ_i = (i+1)λ \;\;, \;\; i≥0$, and death rates ...
1
vote
0answers
19 views

How to distribute two independent rows of bits

Consider two independent rows of 100 bits. The bits are mutually independent and have an equal chance to be 0 or 1. The first row is being read and during that process there is a chance $\epsilon$ ...
0
votes
1answer
16 views

Sum of normal and log-normal independent random variables [closed]

X has a normal distribution, and Y has a log-normal distribution. X and Y are independent random variables. What is the distribution of X+Y? Thank you.
0
votes
3answers
44 views

Finding Variance and Expectation of Boolean Variable

Below is the joint distribution of Boolean random variables X1, X2 and X3. How do I find variance and expectation of X2? I understand that variance is "average of squares of difference from mean ...
0
votes
0answers
19 views

Is there a Continuous Multinomial Distribution??

In Multinomial Distribution, we have \begin{align} f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1\mbox{ and }\dots\mbox{ and }X_k = x_k) \\ \\ & {} = \begin{cases} { \displaystyle ...
0
votes
0answers
11 views

Convergence in Distribution for two Dirichlet distributions

I'm working on a problem and I wanted to get some hints on how to solve it. To me, it seems like showing convergence to distribution but since it's been a while that I've not worked on these types of ...
0
votes
0answers
16 views

Continuous moment generating function but discrete distribution.

I have the following exercise; Suppose that X is a random variable for which the m.g.f. is as follows; $$\psi(t)=\dfrac{e^t}{5}+\dfrac{2e^{4t}}{5}+\dfrac{2e^{8t}}{5}$$ for all $t \in \mathbb{R}.$ ...
1
vote
1answer
24 views

Distribution of product of independent Gaussian random variables

Let $X,Y$ be i.i.d. $\sim N(0,1)$. Then $$\frac{XY}{\sqrt{X^2+Y^2}} \sim N(0,1/4).$$ How can I prove this? I've tried applying the transformation formula, but it hasn't worked out thus far.
0
votes
0answers
14 views

Covariance from Gaussian Mixture Model versus direct calculation

I have a random variable $T$ which can be expressed as follows: $T=c_1P\cdot f+c_2Q$, here $\cdot$ denotes point-wise multiplication. $c_1,c_2$ are scalars and treated as constants. Distribution of ...
-1
votes
0answers
28 views

Probability of getting 3 balls in 10 rooms with 9 people [duplicate]

There is a group of 9 people who visit 10 different rooms together. Each room has 3 balls, and each person has an equal probability of getting a ball. What is the probability that, after visiting all ...
0
votes
0answers
18 views

Mixture of Dirichlet Distributions

I'm working on a problem for Dirichlet distributions and I appreciate if you can give me some hints. Consider two random vectors of size K that are distributed as Dirichlet: $$\vec{Y_1} \sim ...
2
votes
1answer
26 views

When Benford´s Law doesn´t apply - p(d) = (10.5 - d)/(49.5) instead

When Benford´s Law doesn´t apply I would like someone validate (or not) the formula (A) Suppose I pick some book and open it at random. What the probability for first digit page being 1 ? Related ...
-2
votes
1answer
23 views

Poisson distribution help?

The number of car accidents occurring per day on a highway follows a Poisson distribution with mean 1.5. a) Given that at least one accident occurs on another day, find the probability that more than ...
-2
votes
0answers
21 views

Cumulative distribution function and pdf [closed]

Given a situation where the concentration of a pollutant is found to increase with the plant's age according to $C(T) = 10\sqrt{T}$, find the cdf $P(C < c)$. Would it be correct to insert the ...
0
votes
1answer
18 views

PART TWO: Poisson counting process, probability system errors divided in time periods at a certain rate

I've been trying to apply the same knowledge from a previous post, but perhaps my reasoning is wrong. "Errors in a computer surfaces according to a Poisson process with rate 0.4 per day. If there has ...
0
votes
1answer
25 views

Interpreting a condition about CDF

Let F(X) be a strictly increasing CDF which admits a positive density f(x). Is the condition x/F(x) being non-increasing (aka, weakly decreasing) equivalent to saying that F(x) is convex? If not, what ...
1
vote
0answers
40 views

Finding probabilities of a continuous random variable

I have the following continuous random variable density function: $$ f(x) = \begin{cases} \frac14 & if\,0\le x<1 \\ \frac12 & if\,1\le x<2 \\ a & if\,2\le x<4 \\ 0 & ...
0
votes
0answers
3 views

Difficulty to be in first percentile as sample increases.

I was wondering: How can we explain that it is harder to be in the best percentiles as the sample grows ? This question comes from a game called League of Legends, where they introduced a system ...
2
votes
1answer
46 views

Calculate the probability select $k$ blue balls in box

I have a box that contains 10 balls( 2 red balls and 8 blue balls). Probability select each ball is an uniform distribution. An event is defined that selects k balls $(0<k\le 10)$ from the box and ...
1
vote
1answer
39 views

Understanding of the probability using poisson

I have a question about my statistics homework. The question is as follows: At an army base there are X number of soldiers hit by a car. The poisson distribution expaction of this is μ=2. The ...
0
votes
0answers
16 views

Find the PDF of $Y = a*X - b*X^3$ given that $X$ is a uniform random variable on $[0,1]$

Assume that $X$ is a uniform RV on $[0,1]$ and that $a$ and $b$ are both positive. Can also assume that $Y$ is monotonically increasing over its range. I'm trying to find the PDF of $Y$ and am ...
0
votes
1answer
27 views

Poisson counting process, probability system errors divided in time periods at a certain rate

I had some help on a previous post where I learned that The distribution of $X(t)-X(s)$, for $s<t$, is Poisson with rate $\lambda(t-s)$. That is, $$ \mathbb P(X(t)-X(s)=k) = > ...
0
votes
0answers
19 views

How to show analytically the pdf for the minimum of random variables

If $Y_1,Y_2,\ldots,Y_n$ are $i.i.d$ and each $Y_i$ is Generalized Gamma $GG(kn,\lambda)$ distributed. Assuming, the form of $Y_i = Z_i^m$ and each $Z_i^m$ is Gamma distributed, then what will be ...
-1
votes
1answer
19 views

Poisson counting process, probability of arrival of x customers at a certain rate [closed]

I do apologize on beforehand because this post will most definitely be downvoted due to (choose random reason) as I have no clue on how to approach these type of questions. I have been reading my ...
0
votes
1answer
21 views

How to compute selection probability of balls in a range

I have a question about probability that need your help. I assume that I have balls that are numbered from 1 to 100. The probability selection each ball is followed uniform distribution. I divide ...
0
votes
0answers
22 views

A proof involving conditional distributions, Poisson, binomial

Let $X$ be a non-negative integer valued random variable. Let $Y$ be the number of successes in $X$ binomial trials. Prove that, if the distribution of $Y$ and $X\mid (Y=X)$ are identical, then $X$ ...
1
vote
0answers
6 views

distribution of non-central chi random sample

Suppose that $X_1,X_2, \ldots, X_n$ is a random sample from a non-central chi distribution with $1$ degree of freedom. What is the distribution of the sample variance of $X_1,X_2, \ldots, X_n$?
0
votes
0answers
12 views

Working out closed form of shifted poisson distribution

In the article "Bayesian variable selection for Poisson regression with underreported responses" the author defines $t_i^0$ as the number of actual occurences in a study in the $i$th covariate ...
1
vote
1answer
16 views

Two stochastic variables

Say $Y$ is the nummer of accidents by a car, Poisson distributed ($\lambda$). People hit by the car have a probability $p=\frac{1}{2}$ to survive. Let $Z$ be the nummer of people killed by a car ...
0
votes
0answers
8 views

Probability density function with these properties

Consider a family of probability density functions $f$ with parameters $A, B, \mu, \sigma^2$, satisfying the following properties: First, the parameters $A$, $B$ should satisfy $A < B$ and ...
0
votes
0answers
10 views

Compute the expectation of a function of a random vector not knowing the whole distribution

Imagine I have three random variables $X,Y$ and $Z$. I know that $X\sim Y$ which does not imply that $(X,Z)\sim(Y,Z)$ I know the distribution of $(Y,Z)$. So, in a summary: I know the distribution of ...
0
votes
0answers
9 views

Finding the correct distribution

Consider the independent stochasts $X$ and $Y$ that take the values $\pm 1$ with an equal chance (i.e. $1/2$). I came to the following distribution: $f_{X}(x)=(1/2)^{x}*(1/2)^{1-x}=1/2, x=\pm 1$ and ...
0
votes
0answers
11 views

When two random variables that have the same law… Can they be happily exchanges?

Imagine, $X$ and $Y$ are two random variables which have the same law, which we denote by $X\sim Y$. We have then a third random variable $Z$. Can we say that $$(X,Z)\sim (Y,Z)?$$ In what cases is ...
0
votes
0answers
21 views

To prove that induced probability measure indeed defines a probability measure

Given two measurable spaces $(Ω_1, B_1)$ and $(Ω_2, B_2)$, a measurable function T : $Ω_1 → Ω_2$ and P is a probability measure on $(Ω_1, B_1)$. $B_1$ & $B_2$ are respective sigma algebras. The ...
0
votes
2answers
16 views

U,V are two independent random variables each with the uniform distribution on $[0,1]$. What is the $P(V^2 >U>x)$?

$U$, $V$ are two independent random variables each with the uniform distribution on $[0,1]$. Show that $P(V^2>U>x)$ is $1/3 -x +2/3x^{2/3}$ for $0<x<1$. I don't know how to go about ...
0
votes
0answers
21 views

CV Percentage Error with Confidence Variable

I am trying to calculate a confidence variable (CV %) of two numbers where the numbers them selves have a confidence range. Appologies for my sloppy representation, I am somewhat of an equation novice ...
0
votes
0answers
19 views

Gaussian vector multiplied with a matrix is another Gaussian vector: How to show?

Assume that $w$ is a $M$ dimensional random vector, such that: $w \sim N(w|0,\alpha^{-1} I)$. Now I have a $N \times M$ matrix $\Phi$, which is not random. I want show that the vector $Y= \Phi w$ is ...
2
votes
1answer
20 views

Sampling distribution of $\frac{\bar{X}}{S}$

Suppose that I have a random sample $X_1, … ,X_n$ from a $N(0,\sigma^2)$ distribution. What is the distribution of $$\frac{\bar{X}}{S}$$ and what is it's standard deviation? Here $\bar{X}$ is the ...
0
votes
1answer
28 views

Finding $P (X\ge 1)$

If we have the distribution function In part d they want us to find $P (X\ge1)$ So I found the probability distribution. Which is the sum of $f (t)$ for $t \ge 1$ But can we actually find it ...
0
votes
0answers
10 views

How do we call this alternative to CDF: Distribution of outcomes

Let $F(x)$ be a standard cumulative distribution function, associated with a continuous RV. It denotes the probability of random variable $\tilde x$ being on the left-hand-side of $x$: $$ F(x) = ...