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

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13 views

Find mean from geometric PGF

I know that PGF of type 0 geometric random variable is G = p/(1-zs) Now, if I want to find mean, E[X] = d/dz (G) @z=1 = ps/(1-s*z)^2 but according to wikipedia, mean = (1-p)/p and it does not have s ...
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1answer
15 views

Question on finite dimensional distribution of Markov Chain

If $\{ X_{n} \}$ is a Markov Chain and $X_{o} \sim \pi$ (where $\pi$ is the stationary measure), it follows that the MC is identically distributed. I have a question about the finite dimensional ...
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1answer
8 views

What is the maximum of $n$ points with CDF $F$ and PDF $f$?

I read somewhere that the minimum of $n$ points with CDF $F$ and PDF $f$ is $g(y) = n(1-F(y))^{(n-1)}f(y)$ What would the corresponding maximum value of the points be? Also, how do we derive the ...
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2answers
24 views

To use or not Bernoulli trials

I was asked to model the following experiment: Consider the n-th toss of a fair coin, and the event $E$ = '$k$-th toss results in heads'. I find easier to model the experiment using n random ...
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1answer
13 views

Distribution Function - Finding $P(X < 3)$ for a given function

this is the same as this: Distribution Function Of a Random Variable X - Question but that question isnt as clear as I was hoping it was The distribution function of the random variable X is given: ...
3
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1answer
20 views

Deriving master equation for discrete process

Consider a group of $N$ professors, $Y$ of whom are wearing white socks and $X = N − Y$ others who are wearing black socks. On each time step, one professor is chosen at random and he has to put a new ...
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0answers
12 views

Support for a linear combination or transformation of random variables

Let $X, Y \sim iid U(0,1)$ and $c_1, c_2 \in \mathbb{R}$. In the linear combination $Z = c_1X+c_2Y$, we know that the probability density function of $Z$ depends on the relationships of $c_1$ and ...
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0answers
7 views

Is there a analytical formula for Super- and Sub-Poissonian distributions?

I'm currently wrtiting my Bachelors thesis on photon statistics. The way different sources of light can be classified is by Poissonian (coherent light), Super-Poissonian (thermal light) and ...
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0answers
24 views

To determine probability distribution for large $N$ with mean at m [on hold]

To show that the following expression turns to Gaussian for large value of $N$ $$\binom{N}{S}\binom{X-1}{a}\binom{Y-1}{a-1}$$ where X+Y+S=N. To show it shows normal distribution with mean at 'm' and ...
2
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1answer
14 views

Find asymptotic variance MLE heavy tailed distribution

$$\mathbf{X} = \{X_1,X_2,\dots,X_n\}$$ sequence of i.i.d. RV's. Let the distribution of the RV's be defined by $$f(x|\theta)=\frac{\theta}{x^{\theta+1}}, \quad x>1, \quad \theta>1$$ I am ...
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15 views

When Independence $\Rightarrow$ Independence of higher moments (Prob/ Stats)

suppose {$X_n$} is iid. Then, is $X_i$ independent of $X_j^3$ for j≠i? If so, why? Secondly, is $X_i^2$ independent of $X_j^2$ for j≠i? Intuition: yes no If there's a difference, why? (note: ...
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1answer
19 views

Maximum of independent Erlang random variables?

Suppose $Y=\max\{X_1, X_2,\dots,X_N\}$ where all $X_i$ are independent and follows Erlang distribution. I know that extreme value theory deals with maximum of random variables. Can anybody tell me, ...
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2answers
15 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
17 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
24 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
25 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
24 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
46 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
17 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
18 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
82 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
30 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
21 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
11 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
50 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
31 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
17 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
47 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
15 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
14 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
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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
37 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 ...
1
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3answers
45 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
21 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! ...
0
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1answer
24 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
29 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
35 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 ...
0
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1answer
17 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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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
24 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 ...