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

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21 views

Find x, when F(x) is a cdf with given mean and std

I have a small question. For a problem in desicion analysis course, I come to the equation F(x) = 0.8 where X is distributed with mean 50 and standard deviation 10. How can I find x? (my probability ...
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0answers
20 views

Logit Nomal Prior Distribution

$$\mu \sim N(\mu_0,\sigma_0)$$ $$ X_i \sim LN(\mu,\sigma_x)$$ Does anyone know any method for finding the posterior distribution $P(\mu|X)$ or at least any idea of how to estimate it numerically. I ...
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0answers
18 views

Inner product of Gaussians times another multivariate Gaussian derivation help

Assume that: $$p(R|U,V,\alpha) \sim \prod N(R_{ij}|U_i^{T}V_j,\alpha^{-1})$$ $$p(U|\mu_u,\Sigma_u) \sim N(U_{i}|\mu_u,\Sigma_u)$$ $$p(V|\mu_v,\Sigma_v) \sim N(V_{i}|\mu_v,\Sigma_v)$$ Here $R$ is an ...
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1answer
35 views

Generate random numbers from a family of PDFs

For a part of a simulation task, I need to generate (lots of) random numbers from the distribution $$P(E_k | N) = \frac{1}{E_k}\left(\frac{E_k}{k_BT}\right)^{N-1}\frac{1}{(N-2)!} e^{-E_k/k_BT} $$ ...
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1answer
32 views

Given a distribution to generate a set of numbers, what is probability of generating two consecutive numbers whose difference is greater than k?

Suppose I am generating a set of numbers {$x_1$, $x_2$, $x_3$ ... $x_n$} from a given probability distribution $f(x)$. Is it possible to calculate the probability of finding $x_{i+1}-x_i \geq k$, ...
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2answers
36 views

Integration of the product of probability densities

Does a probability density $f(x|\alpha)$ multiplied by another probability density $g(\alpha)$ , where of course both integrate to one, also integrate to one if we integrate with respect to $\alpha$? ...
3
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1answer
67 views

How to find $P(X=r)$ from probability generating function of $X$?

I have a probability generating function $$G_X(s) = \frac{p+ps}{1-s+p+ps}$$ and I need to find $P(X=r)$. How do I get this from the probability generating function? I was thinking about finding ...
2
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1answer
38 views

What is $\Pr[T_a < T_b]$ for independent gamma RVs with same shape

Given independent gamma random variables $T_a, T_b$ with shape $k$ and rates $\lambda_a, \lambda_b$, what is $\Pr[T_a < T_b]$? Estimates are welcome! This question is motivated by the fact that, ...
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1answer
42 views

Scaling a probability distribution function

I have the following PDF that gives the probability of a certain annual wage being drawn: $f(w)=0$ if $w<20000$ $\frac{w-20000}{50000^2}$ if $w \in [20000,70000]$ $\frac{120000-w}{50000^2}$ if ...
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2answers
44 views

negative binomial distribution problem

Find the probability that you find 2 defective tires before 4 good ones. There is a chance of a tire being defective at a rate of 5%. From my understanding with the negative binomial distribution we ...
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0answers
73 views

Expectation of a function of a Binomial random variable

Is it possible to obtain the explicit form of the following expectation, $E\left[\left(\frac{X}{1+X}\right)^n\right]$, where $X$ is a random variable following Binomial distribution with parameters ...
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1answer
74 views

Covariance problem.

The experiment is a three hat experiment with the following probabilities: $\frac15$ for $(1,2,3)$, $(1,3,2)$, $(2,1,3)$ and $(3,2,1)$, and $\frac1{10}$ probability for $(2,3,1)$ and $(3,1,2)$. Find ...
2
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2answers
45 views

Compounding a binomial distribution where the number of observation is binomial?

I am looking for a definition of a binomial distribution where the number of observation is itself binomial. That is: $X \sim binom(N, q)$ When $N \sim binom(n, p)$. Is this a known distribution? And ...
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0answers
144 views

Upper Bound on Supremum of Expected Value

Let $\left( \Omega, F, P\right)$ be a probability space, where $P$ is a probability measure on $\mathbb{X} \subseteq \mathbb{R}^n$, so that $P(\mathbb{X}) = 1$. For all integer $i \geq 1$, consider ...
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1answer
101 views

Uniformly distributed points over the surface of the standard simplex

I would like to generate points that are uniformly distributed over the SURFACE of a standard $k$-simplex ($k$ dimensions, $k+1$ vertices). One way to efficiently generate points that are uniformly ...
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2answers
382 views

Compute Cov(X,Y) while X is the number of 1's and Y is the number of 2's in n dice rolls

Let $X$ be the number of 1's and $Y$ be the number of 2's that occur in $n$ rolls of a fair die. Compute $Cov(X,Y)$. What's wrong with my solution? Here it is: $Cov(X,Y)=E[XY]-E[X]E[Y]$ Compute ...
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1answer
285 views

How to recover the probability mass function from probability generating function?

Would someone please provide me an example of where we take a p.g.f and use it to derive the p.m.f. ? I understand that you were have to take the derivatives of the pmf, which is understandable ...
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2answers
261 views

PDF of X/Y when X, Y are uniformly distributed

The question is as follows: Let $X$ and $Y$ be random variables uniformly distributed on $[0, 1]$. Find the PDF of $Z = Y/X$. I approached it in the following manner: $P(Z < z) = P(Y/X < z) = ...
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2answers
203 views

Modified monty hall problem

Hello how to show the following You are given the choice of 3 doors. Behind one is a car and the other two are goats. You pick a door uniformly at random say 1, and Monty opens another door, say 3 ...
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1answer
44 views

density function of $\frac{1}{\left(X+Y\right)^{2}+1}$.

X and Y are independent continuous random variables with the same density function. Find the density function of $\frac{1}{\left(X+Y\right)^{2}+1}$. I have tried getting the Jacobian where T maps ...
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2answers
104 views

Conditional Probability with marginal densities

X and Y have the joint denstiy: $f(x,y) = 2x+2y-4xy$ for $0< X< 1$ and $0< Y< 1$ and 0 otherwise. . (a) Find The marginal densities of X and Y I got both marginal densities equal to 1 ...
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1answer
44 views

Poisson distribution proof

Looking over an exam and I have no idea how to finish when proving this: Prove that for a Poisson r.v. X, if the parameter $\lambda$ is not fixed and is itself an exponential r.v. with parameter 1, ...
2
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1answer
75 views

Geometric Probability Distribution, Expected Values

Question: Let $X $~ Geometric $(\theta)$, and let $Y = \min(X, 100)$. Compute (a) $E(Y)$ and (b) $E(Y-X)$ I know that the Geometric distribution is $(1-\theta)^{k-1}\theta$ and I also know how to ...
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1answer
30 views

Derivin the Moment Generating Function

Question: Suppose f(x) = 1/10 for 0 < x < 10 and f(x) = 0 elsewhere, compute mx(s) for all s in the Reals This is what I have so far: mx(s) = E(e^(sX)) = integral (from 0 to 10) of ...
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2answers
1k views

Jointly Gaussian uncorrelated random variables are independent [closed]

Let $X,Y$ be jointly normally distributed and uncorrelated. Why are they independent?
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2answers
56 views

What do we know about the pdf of $\bar{X}$

We have $n$ independent random variables $X_i$ all with mean $\theta$ and variance $\sigma^2$. The sample mean is given by $$\bar{X} = \frac{1}{n} \sum\limits_i^n X_i$$ and the means square error is ...
0
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0answers
21 views

Find the Cumulative Distribution Function of F(X) [duplicate]

Supposing that F is the Cumulative Distribution function of some c.r.v. X and that some inverse function F^-1 exists such that F^−1[F(x)] = x for all x ∈ R and F[F^−1(z)] = z for all z ∈ (0, 1) Put ...
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1answer
350 views

How to find first order density of the sample function?

I'm asked to find the first order density of the following process: $x(t)=A\cos(2\pi ft + \theta)$ where, $\theta$ is uniformly distributed over -$\pi$ and $\pi$ and f is a constant. I'm not sure ...
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1answer
280 views

Conditions for convergence of moments given uniform convergence of distribution functions

Setup: Let $S_n = n^{-1} \sum_{i=1}^n X_n$ denote a sample mean and let $S_n^*$ denote a stationary bootstrap re-sample of $S_n$. Let $F_n(x)$ denote the cumulative distribution function of $\sqrt{n} ...
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1answer
86 views

How many students need to be checked to be 90% sure they got a grade of 70..80?

Can you help me on some question i'm stuck on? Let X be a random variable of the average grade of the students in some course. E[X] = 75 ...
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1answer
201 views

Multiple Choice Question Binomial Distribution

Jim didn't study for his math test, and has to guess randomly on 10 multiple choice questions. If each question has 4 choices, what is the probability of gym getting 8 questions correct? I'm ...
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2answers
768 views

Exponential distribution moment generating function to find the mean

With mean = 2 with exponential distribution Calculate $ E(200 + 5Y^2 + 4Y^3) = 432 $ $E(200) = 200 $ $E(5Y^2) = 5E(Y^2) = 5(8) = 40 $ $E(4Y^3) = 4E(Y^3) = 4(48) = 192 $ $E(Y^2) = V(Y) + ...
2
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2answers
58 views

Poisson Distribution for X > y

I'm curious if there is a faster method for solving the following problem: Michael is observing the occurrence of bicycle accidents, and he has determined that B = the number of accidents in one day ...
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1answer
473 views

Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: ...
2
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1answer
36 views

if X and Y are Gauss distributed, what's the distribution of X^2-Y^2?

X and Y are independent random variables with identical Gaussian distribution; for simplicity, the variance shall be 1. What's the distribution of Z=X^2+Y^2? With a plus sign, it would be the ...
2
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1answer
334 views

Show that the nth order statistic is a consistent estimator of a uniform parameter

We assume that our observations come from a uniform $(0,\theta)$ distribution. Can you please check my work on the following? We can derive the distribution function of the maximum of the sample, ...
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2answers
77 views

variance and generating function - probability hat problem

Posted this previously but couldn't comment on it with the temp account i created so: We have a $3$ hat experiment where $(1,2,3), (1,3,2), (2,1,3), (3,2,1)$ have a $\frac{1}{5}$ probability and ...
2
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2answers
87 views

A basic doubt on joint distribution

How to calculate the following probability $P(X \leq x, Y=y)$ where $X$ is a continuous random variable and $Y$ is a discrete random variable. I have been given the distribution of $X$ and ...
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1answer
906 views

cdf of sum of a discrete and continuous random variable

Let U be uniformly distributed on the interval $(0, 2)$ and let V be an independent random variable which has a discrete uniform distribution on $\{0, 1, . . . , n\}$. i.e. $P\{V = i\} =\frac{1}{n+1}$ ...
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2answers
57 views

Limiting case of Binomial(n,p)/n?

Let the random variable $X$ have distribution $X \sim \text{Binomial}(n,p)$. Let $Y = X/n$. What is the limiting distribution of $Y$, as $n \to \infty$? Does it have a simple distribution? Of ...
0
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1answer
211 views

probability dreidel game problem

Reuven and Shimon play a dreidel game. In each round, each of them rolls the dreidel.(dreidel can fall on G,N,H,P only in equal probability) If one of them gets a G and the other not, he wins; if both ...
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1answer
54 views

Probability of an Event defined by two continuous random variables

I'm having trouble solving this word problem. I have the answer, but do not know how to get there. An electronic gadget employs two integrated circuit chips: a signal processing chip and a power ...
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1answer
162 views

Books on Probability Theory (for an engineer)

I am an electrical engineering student who deals with probability and stochastic processes for communications systems, but I find that most engineering texts on probability and stochastic processes ...
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2answers
131 views

Addition of probabilities and gambler's fallacy

Say you have a 1 in 6 chance of winning a card game. The more times you play, the higher the odds of you winning. $$P(\text{win over 1 trial}) = 1/6 \\ P(\text{win over 2 trials}) = 1/6 + 1/6 \\ ... ...
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1answer
64 views

Cumulative distribution function changes variable

I am really at a loss for what this question is even asking. Could someone please explain it to me? Suppose continuous random variable X has the cumulative distribution function F(x). Find the CDF ...
2
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1answer
70 views

Median and Mean of Sum of Two Exponentials

I have a cumulative distribution function: $$G(x) = -ae^{-xb} - ce^{-xd}+h$$ The associated probability density function is: $$g(x) = abe^{-xb} + cde^{-xd}$$ My problem concerns $x\ge 0, X \in R$. I ...
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1answer
111 views

Derived Distributions: PDF of -ln|X| [answered]

I am studying questions for a probability exam. I am stuck on derived distributions. One of my textbook's questions asks: If $X$ is a random variable uniformly distributed between $-1$ and $1$, find ...
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2answers
168 views

Chi-square distribution Verification

I know there is one answer for this question, but I just changed my question a little bit...so please read it By Definition, Let ν be a positive integer. A random variable Y is said to have a ...
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1answer
92 views

Why is it so easy to marginalize a multivariate random distribution?

From wikipedia: To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to ...
0
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1answer
45 views

proving unbiasedness of an estimator

Question given independent random variable $X_{1},X_{2},...,X_{n}$ from a geometric distribution with parameter $p$. we have an estimator for $p$, mainly $T=Y/n$ where Y is number of $i$ that ...