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

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Matching Gamma Statistics with Poison Statistics

Confidence intervals with Poisson distribution would be greatly helped by matching an equivalent gamma distribution. Can someone lay out how to match a Gamma Distribution to a poisson distribution? ...
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5 views

Sampling from the Marshall-Olkin Copula

I know the algorithm, just cannot understand why it works. There are 2 processes with running times $T_1$ and $T_2$ and 3 exponentially distributed shocks $\tau_1,\tau_2,\tau_{12}$ each with ...
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8 views

Relation between minimum and sum of two random variable

I am interested in finding a relation that involves two independant random variables, that could be used to describe the sum of these, or the minimum of these. For example, regarding the sum, we know ...
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10 views

Maximum of Correlated random variables

I am trying to find the CDF $Z = \max(X_1,X_2,\dots,X_N)$ and in my case $X_i$ are correlated. Is there any transfer domain or one to one function where I can derive the CDF and invert back to current ...
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1answer
8 views

Equality of two binomial parameters

I am having a problem with this problem of Ross. Can anyone please help me out. In a famous experiment to determine the efficacy of aspirin in preventing heart attacks, 22,000 healthy middle-aged men ...
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11 views

Help me understand how finding related distributions work

$$G(a)=\frac{6400}{a^2}$$ So this is the question, and I know well to answer any type of question like this. Here's how I do? (i) $$F(t)=\int_{1/2}^{T} \frac {1}{2t^3} dt = ...
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15 views

How to solve using Jacobian?

I am looking for a way to find the joint pdf of vector $Z=[Z_1,Z_2,Z_3,Z_4]$ where $Z_1= a_1 X_1^2 + a_2X_1Y_1+ a_3 X_1Y_2 + a_4Y_1^2 + a_5Y_2^2$ $Z_2= b_1 X_1^2 + b_2X_1Y_1+ b_3 X_1Y_2 + b_4Y_1^2 + ...
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11 views

A reference for a Gaussian inequality {$\mathbb{E} \max_i X_i$)

I am looking for a reference to cite, for the following "folklore" asymptotic behaviour of the maximum of $n$ independent Gaussian real-valued random variables $X_1,\dots, X_n\sim ...
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1answer
14 views

Given a data set of random values in MATLAB, how do I plot its PDF?

Suppose I have the following code in MATLAB: ...
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1answer
17 views

uniqueness of joint probability mass function given the marginals and the covariance

Let X and Y be two nonnegative, integer-valued random variables. Is there a way to find the joint probability mass function, i.e. $$ \mathbb{P}(X= k, Y= h) $$ for some $k,h\geq 0$, given the ...
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1answer
27 views

How many will not be selected in repeated tries?

Suppose I have $25$ uniquely identifiable objects, i.e., I know which is which once it has been selected (but they are not distinguishable in the selection process). I select $5$ objects at random, ...
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19 views

Find $E[Z_1 | aZ_1 + bZ_2]$

Let's $Z_1,Z_2$ be a random variable such that $EZ_1^2 < \infty$ and $EZ_2^2 < \infty$. Find $E[Z_1 | aZ_1 + bZ_2]$ where $a,b \in \mathbb{R}$. We don't know what is distribution of $Z_1$ and ...
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1answer
30 views

Conditional probability problem and Alias Method

I hopefully someone can help me with this problem of conditional probability: "A disk server receives requests from many client machines and requires 10 milliseconds to respond to each request. The ...
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22 views

how to related a weakly convergent random variable with its k-th moment

Let $\{X_n\}$ be independent random sequence with zero mean and unit variance. Suppose $$S_n:=\sum_{m=1}^n \frac{X_m}{\sqrt{n}} \Rightarrow X\sim \mathcal{N}(0,1)$$ holds. (Here "$\Rightarrow$" ...
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30 views

necessary conditions for these conditionals to be consistent with some joint distribution

Let $A$, $B$, and $C$ be random variables taking discrete values in the set $\{0,1\}$. I'm trying to find necessary conditions such that the conditional distributions $$X\mid Y,\,Y\mid Z,\,Z\mid X$$ ...
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15 views

Probability distribtuions [on hold]

A 10 metre by 10 metre plot of land is divided into 100 equally sized squares. Suppose that 300 seeds are randomly scattered on the plot of land. Use a suitable approximation to find the probability ...
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15 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
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2answers
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Finding a constant from a continuous distribution

$X$ is a continuous random variable with PDF $$f(x) = c\theta^{|x|} \quad \text{ for } -\infty<x<\infty,$$ where $0<\theta<1$ is a parameter and $c$ is a constant. Derive and expression ...
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16 views

Function of Nakagami Distribution

Does anyone know what the distribution of the sum of squared Nakagami is? $$\sum_i^n X_i^2$$ $$X_i\sim \text{Complex Nakagami-m }$$ Is the distribution Erlang? Is the distribution the same as ...
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1answer
8 views

How do you compute the PDF of a function of 2 random variables that is not a sum?

If you have a random variable U(X,Y) that is a function of two other random variables X and Y such that $U(X,Y)=X+Y$ and you know the PDFs of X and Y are defined to be exponential such that $f(t) ...
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1answer
8 views

How to interpret this variance

If I have a probability measure defined by $P( \Omega ) = \int_{\Omega} (1-a) \delta(x) + a \delta(x-a^2) dx,$ then I noticed that the variance is given by $a^5(1-a)$. This is somewhat strange, cause ...
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how to find expected value with toys? [duplicate]

A couple years ago, Burger King was giving a Dragon Ball Z toy in every kids meal. There were 6 unique toys that you could collect. Lets say I randomly selected a toy in each kids meal. What is the ...
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15 views

What is the magnitude of Complex random variable Gaussian Case?

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
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1answer
23 views

marginal pdf of a exponential distribution

Problem Let $X$ have the pdf $f(x)=e^{-x}$, $x>0$ and $Y$ have the pdf $f(y)=e^{-y}$, $y>0$. Assume that X and Y are independent Find the pdf of $U=X+Y$? Solution Since X and Y are indep. it ...
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2answers
29 views

For a non-negative absolutely continuous random variable $X$, with distribution $F$. Why is $\lim_{t\rightarrow \infty}t(1-F(t))=0$?

So I am given a non-negative absolutely continuous random variable $X$ with distribution $F$, and density $p_X$. I am given the definition of expectation using simple functions and the survival ...
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0answers
18 views

statistics uniform distribution [on hold]

Dominic released his rabbit to roam on the lawn; some time later, it returned and so he continued doing that daily. Over time, he found a pattern of the time T (in hours) Rabbit stayed outside. If ...
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1answer
21 views

A question of joint CDF

I am confused about how to use the joint probabilities to find the joint CDF.
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11 views

Probability that one random variable is greater than or equal to another

Assume $X$ and $Y$ are i.i.d. with exponential distribution with parameter $\lambda = 1$ (the probability density functions $p_X (x) = e^{-x}$ and $p_Y (y) = e^{-x}$ in $[0, +\infty)$, $0$ otherwise). ...
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Unable to follow notation and meaning of probability distribution for binary time series

I am unable to understand concepts related to the probability distribution of binary time series. [Mathematics is not my background]. This is from the book Binary time series by Benjamin Kedem, vol ...
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1answer
13 views

Probability of Normal Distribution

Let's say that 10 sumo wrestlers were to squeeze into an elevator that could only hold a max capacity of 2300 pounds. Let's say that the weight of the sumo wrestlers is normally distributed with a ...
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1answer
20 views

Elementary Probability: Expected Value

I must say, first, that this question IS a homework assignment and I do not wish an answer here, for I already posssess it. I want to know if there is a general procedure of simplification in this ...
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1answer
16 views

A question of Joint PDF

I have not idea about part a. I know I need to prove the integration of f(x,y)=1, but how should I deal with the range of x and y.
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10 views

Deriving a simple PDF

I am looking for deriving the pdf of $Z$ where $Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$, where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
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1answer
28 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived ...
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1answer
16 views

Conditional Expectation of Binomial Given $X \leq x$

Are there any neat formulas to reduce something like $\sum_{i=0}^{x} i \binom{n}{i} p^i (1-p)^{n-i}$ where $x<n$? This would be proportional to $\mathbb{E}(X\leq x)$ where $X$~$\text{Bin}(n,p)$. ...
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1answer
12 views

Joint CDF from conditional cdf

I would like to derive an expression of the following joint CDF $P[X \leq x,Y \leq y]$ based on the conditional CDF $P[X \leq x | Y=y]$ and the pdf $P[Y=y]$ that are considered to be known. I get a ...
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1answer
33 views

Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$

Let $m$ be an arbitrary value in $Z_n$, where n is RSA modulo (n=p.q, where p and q are large primes). Then have: $r_2=r_1 . m$, where $r_1$ is a value chosen uniformly at random : $r_1\in Z^*_n$. ...
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22 views

The Joint PDF question [on hold]

Can someone help me do this series of question? I really need help and I have no idea about it. I have no idea of how to deal with the range of x and y.
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2answers
17 views

Probability Distribution Function?

An urn contains 8 green balls and 17 yellow balls. A ball is drawn from the urn and its color is noted and then the ball is placed back in the urn. 5 balls are drawn this way. Let $X$ denote the ...
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1answer
31 views

Negative binomial with conditional probability

Let X be a random variable that follows a negative binomial distribution: NB(r=4, p=0.4) Calculate P(X = 8 | x > 6) I know how to calculate P(X = 8): $$ \binom{7}{3} \cdot (1 - 0.4)^{7-3} \cdot ...
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1answer
21 views

How to set up problem involving Poisson RV

Consider an example where customers entering a store is a Poisson random variable with $\lambda=15$. How do you find the probability that 100 or fewer people will walk into the store in any five-day ...
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1answer
31 views

Prove that $ \lim_{n \to \infty} \frac{\Phi(- \sqrt{n})}{f(\sqrt{n})} = 1$.

Let $X$ be standard normal random variable. Let $\Phi$ be a distribution function of $X$ and $f$ density function of $X$. Prove that $$ \lim_{n \to \infty} \frac{\Phi(- \sqrt{n})}{f(\sqrt{n})} = 1$$. ...
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22 views

Easy question: Multiple random variables vs. product of probability spaces

I never had a course in probability theory and the definitions we work with are quite informal, so I am a little bit confused about the difference between "multiple" random variables and the notion of ...
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25 views

CDF of ratio of Gamma distribution with different parameters

Let $X$ be gamma distributed random variable with parameters $a$ and $b$. Let $W$ be gamma distributed random variable with parameters $c$ and $d$, such that \begin{equation} f_X(x) = ...
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How to find the distribution of $W = (X_1^2 + X_2^2) / 4$, where both $X_i$ are iid with given moment generating function?

$X_i$s are i.i.d random variables. (The number of random variables is not specified.) And m.g.f of X is $M(t) = \exp[2(t^2)]$. How can I get the distribution of new random variable $W = (X_1^2 ...
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1answer
21 views

Using the binomial distribution as the distribution for a sum of Bernoulli random variables?

Knowing that the sum of $n$ independent Bernoulli random variables with parameter $p$ ($p \in (0,1)$) has a binomial distribution $Bin(n,p)$, how can I use the Central Limit Theorem (or any other ...
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20 views

joint distribution of two arbitrary distributions?

F = S + E where S: start time and E: execution time, which are arbitrary probability distributions. S and E are discrete and independent.F is finish time of a task which starts in random start time ...
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36 views

How to demonstrate a particular functional equation solution

In order to find a prior probability distribution I have to solve the following functional equation: $$af\left(\frac{a\theta}{1-\theta-a\theta}\right)=(1-\theta+a\theta)^2f(\theta)$$ the solution of ...
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1answer
11 views

Limiting random variable

My question is about limiting r.v. Suppose, we have a sequence of r.v.s. $\{X_n\}$. And we know that $\liminf X_n=-\infty$ and $\limsup X_n =\infty$ almost surely. What can we say about $\lim X_n$. ...
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1answer
38 views

Applying the Poisson Distribution to problems

The number of traffic accidents at a certain intersection is thought to be well modeled by a Poisson process with a mean of $3$ accidents per year. Find the probability that more than one year ...