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

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Maximum of independent Erlang random variables follows which extreme value distribution?

Suppose $Y=\max\{X_1, X_2,\dots,X_N\}$ where all $X_i$ are independent and follows gamma distribution. I know that extreme value theory deals with maximum of random variables. Can anybody tell me, ...
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2answers
12 views

Conditional probability of a Joint distribution

Let $(X,Y)$ have joint density $f(x,y)=e^{-y}$ , for $0<x<y$, and $f(x,y)=0$ elsewhere. What is $f_{X\mid Y} (x,y)$ for $0<x<y$? I think that the answer is $1/y$, however, I am having ...
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1answer
13 views

How to compute Binomial Distribution?

A basketball player scores a point in a free throw with 80% probability. Probability is independent of the result of the previous throw. 1) Given 5 free throws, find the probability distribution of ...
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1answer
21 views

Question of Poisson Distribution

I'm still confused about how to apply the Poisson Distribution, could you help me to explain how to solve the following problem? A company department takes on average 2 new employees per year. New ...
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1answer
19 views

Basic finite dimensional distribution question

I'm having trouble wrapping my head around the basic idea of a finite dimensional distribution. Suppose $(\Omega, \Bbb P, \mathcal{F})$ is a probability space. Let $(X_{t})_{t \geq 0}$ be a ...
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1answer
22 views

How to calculate the probability distribution function (PDF) and the cumulative distribution function (CDF)?

Sorry I'm a novice to both functions and just didn't get a clue how to solve this problem (having been reading the theories for the whole day but still ...) The problem is: We have now two investment ...
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0answers
12 views

Independence given conditional pdf of X|Y and marginal pdf of Y

I'm a new poster on Stack Exchange though I've been using this site as a useful resource for a while now. This is a homework question that I wish to get clarification for: Given the random variables ...
6
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1answer
42 views

Trying to understand the behaviour of i.i.d.

In a course called introduction to probability theorem we are covering now i.i.d. (independent and identically distributed random variables). I already know when two variables are independent: $X, Y$ ...
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0answers
14 views

When is a coupling ''natural''?

The definition of coupling is written below. In some articles, I found the term "natural coupling". When is a coupling said to be ''natural''? Definition of coupling between two random variables: Let ...
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1answer
15 views

Distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion

I am trying to find the distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion. One (very hand-wavey) way is to assume a priori that it is Normally distributed. Then one can ...
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2answers
78 views

Clever way of finding $\int_0^\infty x\Phi(x)\phi(x)dx$

Suppose that $\Phi$ and $\phi$ are the Standard Normal c.d.f and p.d.f. respectively. Then, evaluate $$\int_0^\infty x\Phi(x)\phi(x)dx$$ There is no use of my trying to show my approach because ...
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1answer
28 views

Joint probability distribution of AB given A=X/Y, B=Y, the distributions of X and Y

I have recently been brushing up on some statistics in preparation for further study and I have encountered this question that has stumped me quite a bit: Find the joint probability distribution of A ...
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1answer
20 views

Decision-making with random term

Consider the following situation. There are multiple options to choose from based on an attribute related to those options. For example: ...
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1answer
34 views

Non-standard question about random variables

I am not sure which subbranch of mathematics this is, so I cannot give a precise tag. I am doing research, and this suddenly popped out of no where. So, please hear me out. $x$ is a variable that ...
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1answer
10 views

new bounds for transformed random variable

Let $Y \sim U\left ( 0,1 \right)$, I have already determined the new pdf for the transformation $Z=Y^2$. I used the cdf technique for this. So the new pdf for $Z=Y^2$ is $f_Z(z) = ...
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0answers
17 views

Integration using t distribuion [on hold]

integrating using student t distribution In the above question, as per the solution posted I don't get how the final answer is $\frac{\pi}{2}$ because $ \gamma(1) = 1 $ and $\gamma(1/2) = ...
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2answers
48 views

Find the distribution - coin is tossed three times

A fair coin is tossed three times. Let $X$ be the number of heads that turn up on the first two tosses and $Y$ the number of heads that turn up on the third toss. Give the distribution of $X$, $Y$, $X ...
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1answer
30 views

Find distribution and the expected value of final grade [on hold]

A performance is graded independently by three experts (the possible grades are as follows: 1, 2, 3, 4, 5), and then the highest and the lowest mark are crossed out. The remaininggrade is the final ...
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1answer
16 views

Prove that a symmetric distribution has zero skewness

Prove that a symmetric distribution has zero skewness. Okay so the question states : First prove that a distribution symmetric about a point a, has mean a. I found an answer on how to prove this ...
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1answer
41 views

Jee Main 2015 Question. Probabilty

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is: (1) $22 \times(\frac{1}{3})^{11}$ (2) $\frac{55}{3} \times ...
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0answers
14 views

How to draw marginal density function using R?

suppose $f(x,y) = cxy^2$ is the joint pdf of $X$ and $Y$. $0\le x\ ,\ y\le 2$. Q1: what value must $c$ have for this to be a pdf? Q2: what is the marginal distributions of $X$ and $Y$. ...
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1answer
22 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-1} \mathbf P(X=z) ...
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0answers
13 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 ...
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1answer
20 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, ...
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3answers
36 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 = ...
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0answers
12 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 ...
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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 ...
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3answers
44 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 ...
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1answer
20 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 $\|(X_1,Y_1)-(X_2,Y_2)\|$?
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0answers
27 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! ...
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1answer
22 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 ...
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2answers
27 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 ...
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1answer
34 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 $t$, ...
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0answers
42 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.
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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 ...
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1answer
25 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 ...
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1answer
16 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 ...
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12 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 ...
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1answer
32 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 ...
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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 ...
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1answer
22 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
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1answer
44 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 ...
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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 ...
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2answers
32 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 ...
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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 ...
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0answers
20 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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0answers
35 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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2answers
37 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 ...
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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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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 $E[X+Y]$