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

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0answers
8 views

How to draw marginal density function using R?

suppose f(x,y) = cxy^2 is the joint pdf of X and Y. 0<=x,y <=2. Q1: what value must c have for this to be a pdf? Q2: what is the marginal distributions of X and Y. Q3: draw a graph of each ...
0
votes
1answer
14 views

Derivation of negative binomial distribution

Let $X, Y$ be geometric distribution where $ \mathbf P(X=k) = \mathbf P(Y=k) = (1-p)p^{k-1}$ for $k = 1, 2, 3...$ Using the convolution formula: $$\mathbf P(Z=z)=\sum_{n=1}^{z} \mathbf P(X=z) ...
0
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0answers
5 views

Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...
1
vote
1answer
12 views

Distribution of random variables (normal and standard normal)

Suppose that $X_i \sim N(\mu, \sigma^2)$ for $i = 1, \ldots, n$ and that $Z_i \sim N(0,1)$ where all of the random variables are independent. Denote $s^2_Z$ as the sample variance of $Z_1 , \ldots, ...
1
vote
3answers
34 views

Hello expected output (probability question)

I am working on a probability problem I tried finding the total net productivity days based on the amount of machines the factory has, so if there was 1 machine, there will be 29 days * 1 machine = ...
1
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0answers
10 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of ...
0
votes
1answer
16 views

Selection of Distribution model

An expressed parcel delivery company offers a First Class service for which it is promised that 80% of all parcels are delivered within 24 hours of dispatch. It is suspected that the true successful ...
1
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3answers
32 views

Expected no of balls to select before a certain type of ball comes

There are w white balls and r red balls in a box, to find the expected no of balls to pick before we get a red ball? $$\qquad$$ What I have tried is, Let $ X_k $ denote that k no of white balls have ...
1
vote
1answer
18 views

distribution of distance between two points whose coordinates are normal random variables

let there be two random variables $(X_1,Y_1)$ and $(X_2,Y_2)$, where $X_1\sim N(m_1,s)$, $X_2\sim N(m2,s)$, $Y_1\sim N(n,t)$, $Y_2\sim N(n,t)$. What is the distribution of $\|(X_1,Y_1)-(X_2,Y_2)\|$?
0
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0answers
25 views

Conditional distribution of two binomials which both depend on a third

I have a question that I'm having some trouble with, but which I believe might have a fairly straightforward answer. I'd really appreciate it if someone could help point me in the right direction! ...
0
votes
1answer
17 views

computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
0
votes
2answers
21 views

Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
1
vote
1answer
30 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, ...
-1
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0answers
30 views

Probability_distribution [on hold]

Three points are chosen at random on the circumference of a circle. Find the probability that they all lie on the same semicircle, using random numbers generated from a uniform distribution.
0
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0answers
6 views

Efficient random sample from Markov chain with known states at two times

Assume a 2-state Markov chain with known transition matrix. Suppose I know, for example, that the chain is in state 1 at time 1, and is also in state 0 at time 10. I want to sample randomly from the ...
0
votes
1answer
23 views

Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...
0
votes
1answer
10 views

Bounds-negative binomial distribution

Suppose $Y=\sum_{i=1}^{n} X_{i}$ where each $X_{i}$ is an independently and identically distributed geometric random variable with success parameter $p$, so that $Y$ has a negative binomial ...
0
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0answers
11 views

Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
-1
votes
1answer
31 views

Uniformly distributed independent random Variables [on hold]

Let X and Y be independent random variables each uniformly distributed on (0,1). Find $P(Y\geq X | Y\geq \frac{1}{2})$. The answer is $\frac{3}{4}$ But I don't know how they got it :( Please help as I ...
1
vote
2answers
23 views

Find the distribution function of bivariate distribution

Find the distribution function of $$f_{X,Y}(x,y)=\begin{cases} e^{-y}, & \text{if $0< x<y < \infty$} \\ 0, & \text{ otherwise} \end{cases}$$ Trial : According to my calculation ...
0
votes
1answer
20 views

Joint density calculation

Let $X$ have a (standard) normal distribution; with zero mean and unit variance. Let $Y=WX$ where $\mathsf P(W=1) = \mathsf P(W=-1) = \tfrac{1}{2}$. What are the joint and conditional probability ...
2
votes
1answer
43 views

Probability that the proportion of a shorter segment with relation to the longer one is less than $\dfrac{1}{4}$

The problem is as follows. We randomly pick a point on a segment line of lenght L. What is the probability that the quotient of the shorter segment with relation to the longer one is less than ...
0
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0answers
8 views

Cumulative Distribution Function from mean and stdev [on hold]

I found the following equation in some source code to find the values of the cumulative distribution function at several points: pn(z) = 1/(1+e^(-1.59z - 0.727z^3)) where z = (x-mu)/sigma where mu ...
0
votes
2answers
30 views

Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...
3
votes
1answer
27 views

Probability Joint Density Question [on hold]

Suppose $(X, Y )$ is uniformly distributed over the set $\{(x, y) : 0 < y + x < 2, 0 < x < 2\}$. Find the joint density of $(X,Y)$ and marginal density of $F_Y(y)$. I am having a tough ...
0
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0answers
19 views

Is there a way to derive the skewness formulae for different distributions?

I would like to know if there is a way to derive the formula to calculate skewness for different distributions, as they are not included on the formula sheet in the coming exams. For example, ...
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votes
0answers
34 views

there are 100 students in a math class, 36 are male and study pure math, [on hold]

9 are male and not studying pure math, 42 are female and studying pure math, 13 are female and not studying pure math, use the data to deduce probabilities concerning a student drawn at random
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votes
2answers
36 views

Exponential distribution of random variable [on hold]

Random variable $X$ has probability density function $g(x)=\frac{3}{7}x^2\mathbf{1}_{[1,2]}$. Is there a function $F: \mathbb{R}\to\mathbb{R}$ for which $F(X)$ has an exponential distribution with ...
0
votes
3answers
21 views

Confused by (cumulative) distribution function question…

$P(0<=X<1)$ if $X$ is a random variable having a distribution function: $F(x)=$ {($0, x<0$), ($1/3, 0<=x<1$), ($2/3, 1<=x<2$), ($1, x>=2)$} (hope that makes sense) But if $x$ ...
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votes
0answers
18 views

Find the PGF of two independent binomial random variables [on hold]

Let $X$ and $Y$ be independent binomial random variables with parameters $(n_1,p_1)$ and $(n_2,p_2)$ respectively. Find the PGF $\phi_{X+Y}(z)$, find the expectation $E[X+Y]$
0
votes
1answer
25 views

simplify the division of popular probability density function

This is my first question in Mathematics on Stack Exchange. Please forgive that this is a none sense question... Question I'd like to know a simple form of the division of popular probability ...
0
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0answers
12 views

Probability density function for a PDE with random inputs

I am looking for a general method or alternatively few textbook examples of deriving a probability density function for a solution of partial differential equation with random inputs in the equation ...
0
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0answers
24 views

CDF of sum of 3 independent discrete uniform random variables on {1,2,…,n}

What is an approximate closed formula for this probability, with a derivation: p(k,n) is the probability, that among $n$ PC discs and $k$ errors in sum on them, there will be at least $1$ disc ...
0
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1answer
32 views

Calculating probabilities for complex random variables

I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable $Z$ as: ...
0
votes
1answer
43 views

Prove that a function is decreasing

Let $\left(\,c_m\,\right)_{m \in \mathbb{N}}$ be some coefficients which are all positive natural, $c_0=1$, and they are increasing in $m$. Define $$ f(y) = \frac{\sum\limits_{m=0} c_m \, \, ( y ...
2
votes
1answer
33 views

Computer Component with Gamma Distribution?

I comes to a question of one old-exam as follows: if the life of one computer component (in year) has Gamma Distribution (if I translate correctly) with ...
1
vote
2answers
31 views

Two company and probability example?

I ran into a problem that seems strange to me. Two companies A,B produce a device that with probability $0.05$ and $0.01$ are broken. if we buy two devices produced by one company with equal ...
0
votes
0answers
29 views

Stochastic dominance of Binomial and Poission

In order to investigate the size of the cluster of a given vetex in a random graph I need to use a fact about stochastic dominance that I don't know how to prove. Namely, I am looking for a proof of ...
0
votes
0answers
21 views

Difference in Chebyshev inequality.

I am following a lecture, that provides the following formula of the Chebyshev inequality: $$P\left \{ \left | \frac{S_n}{n} - p \right | > \varepsilon \right \} \leqslant \frac{1}{4n\varepsilon ...
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votes
0answers
25 views

Expected value problem

Imagine the following scenario: Several players contribute coins to a pot in a random order and without knowing what other players have contributed. When every player has added his stake, the coins ...
1
vote
1answer
34 views

Distribution of P[Y=n] = P[n-1<X<n] for X exponentially distributed

From an assignment, we have "Let X be an exponentially distributed random variable with probability density function. $f(x) = λe^{−λx}$, for $x > 0$" I've worked out that for $P[Y=n] = P[n-1 < ...
1
vote
0answers
40 views

Question about $M/GI/ \infty $ queue

Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...
0
votes
1answer
23 views

What distribution models number of trials needed for given number of successes and success rate?

Case scenario: a retro-virus infects a healthy cell. The virus programs the cell to brew little viruses, at a rate of 0.5 per-sec, until finally the cell bursts when the number of virus inside it is ...
-3
votes
1answer
33 views

Probability nad Statistics [on hold]

A jar contains 5 blue dice and 2 red dice, three dice are drawn at random from the jar then they are rolled. a)What is the probability of getting the sum of 14 from rolling 3 dice? b)What is the ...
1
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0answers
25 views

Simulating r.v.'s from a joint density by rejection sampling in R. Continued

I wish to sample variables $v$ and $w$ from the joint density $$(v+w)e^{-\frac{(v+w)^{2}}{2x_{0}}-2\mu v-(\mu -\lambda )w},$$ where $x_0$, $\mu$ and $\lambda$ can be seen as positive constant. Since ...
0
votes
0answers
27 views

Applying chain rule in probability?

Let $X,Y$ be random variables with distribution functions $F_X(x)$, $F_Y(y)$. Let $W(u,v)=max\{0,u+v-1\}$. why can we take the following limits "inside" $W$? $lim_{(x,y)\to ...
1
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0answers
25 views

Prove that the variance of a discrete random variable increases with a parameter

I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$. Let ...
0
votes
2answers
33 views

Does convexity of the distribution function imply convexity of the density function? [on hold]

If a distribution function $F$ is convex, such that $$ \frac{\partial^2}{\partial a^2}F(x,a)\ge 0 $$ does this then imply that it density $f$ is also convex, such that $$ \frac{\partial^2}{\partial ...
0
votes
1answer
24 views

Is the support of the Gaussian finite or infinite?

Considering that as $x \to \pm \infty$ ; $e^{-\frac{x^2}{2}} \to 0$, is the support finite or infinite? A simple enough question, but enough to make me scratch my head. I feel that it's almost a ...
0
votes
0answers
35 views

Probability density function for distance between two points.

Two points are chosen randomly inside a circle (and even on the circumference) with radius $r$ What is the probability density function of the distance between the points? I would be very grateful.