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

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2
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1answer
42 views

Why Characteristic function is given such a name?

I've come to know that, For random variable $X$ ,with a Probability mass function P, $\phi_X(t)$ defined by : $\phi_X(t) : \mathbb R \to \mathbb C$ $\phi_X(t) = E(e^{itX}) =E[\cos tX + i\sin tX]$ ...
1
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0answers
12 views

Calculating the probability distribution function of a realization of a random process

I have a realization of a continuous random process, $y = f(t)$, a function of time ($t$). I am trying to calculate $P(y = Y_0)$, the probability distribution function of $f(t)$. Am I right in saying ...
0
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0answers
13 views
+50

Transforming a categorical distribution by repeating trials and taking a plurality

Suppose you have a K-sided, weighted die. This is represented by a categorical distribution. Now, let's say you roll the die N times, and then pick a "winner" by choosing whichever outcome has a ...
0
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0answers
10 views

How do we get parameters from distribution?

I've estimated the distributions for options prices using $F(X_n) = e^{rT}\frac{\delta C}{\delta X} + 1=e^{rT}[\frac{C_{n+1}-C_{n-1}}{X_{n+1}-X_{n-1}}]+1$ $F(X_n) = e^{rT}\frac{\delta^2 C}{\delta X^...
1
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1answer
41 views

Help understanding convolutions for probability?

I have been trying to do some problems in probability that use convolutions but there has not been much of an explanation of what a convolution is or the purpose of using a convolution. For example ...
1
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0answers
28 views

Simplifying Multiple Integral for Compound Probability Density Function

Are there any ways to simplify this multiple integral? $$ \hat{f}\left(\left.y\right|\alpha\right)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\hat{f}\left(\left.y\right|\theta_{1}\right)\hat{...
0
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0answers
16 views

How to derive the cumulative density function of $X>0$ from the characteristic function of $Y=X^2$?

Which is the relation (can the 1st be derived knowing the 2nd) between the cumulative density function of positive rv $X>0$$CDF_X(x)=Prob(C<=x)$ and the characteristic function of $Y=X^2$: $CF_{...
0
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0answers
21 views

Continuous Joint conditional probability

I am have been trying to do the following problem. Suppose X, Y are two continuous random variables with joint probability density function: $$f_{XY}(x,y)= \begin{cases} 12xy(1-x) & 0 <x,y<...
0
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0answers
16 views

Maxwell–Boltzmann distribution average speed and second-order moment

Someone can give me a link where I can find the solution step by step of the following integrals. Otherwise,if there is someone so kind to solve them. $$dN(v)=4\pi N(\frac{m}{2\pi kT})^{\frac{3}{2}}...
0
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1answer
19 views

Discrete probability distribution: is the game fair?

A and B play a game by rolling 2 dice. A gets 5 points if he rolls a double (both dice the same), otherwise he loses 1 point. Is the game fair? What is the expectation gain or loss for A? Is ...
0
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1answer
38 views

Find $\mathbb{P}(X>Y)$ given the distribution

Random variable (X,Y) has a uniform distribution over a triangle with vertices at $(1,0),(0,1),(-1,0)$. Find $P(X>Y)$ obviously it is going to be a double integral the answer i have in my answer ...
1
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0answers
41 views

Probability that an arbitrary element of a field has a specific structure.

This question is related to : http://crypto.stackexchange.com/questions/37351/encoding-an-element-in-r-rhr-way that I asked couple of weeks ago. The difference is that I did not take the collision ...
0
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0answers
22 views

Hypothesis testing for the correlation coefficient

My question is related to the correlation between random variables X and Y, where $(X,Y)$ is bivariate normal. My understanding is as follows. The correlation coefficient is $\rho=\dfrac{\...
1
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2answers
44 views

Need help finding joint distribution of uniform and exponential

Let X be an exponential random variable with parameter λ and Y be a uniform random variable on [0,1] independent of X. Find the probability density function of X + Y. Now I have computed this ...
1
vote
1answer
34 views

How we can show $\mathbb{E}[T]=0$ and $\operatorname{Var}(T)=\frac{n}{n-2}$.

I need help with this question. Let $Z\sim N(0,1)$ and $Y\sim X^2_{(n)}$ be independent variables, and define$$T\stackrel{\rm def}{=}\frac{Z}{\sqrt{\frac{Y}{n}}}.$$ Prove that $\mathbb{E}[T]=0$...
0
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3answers
45 views

Inequality for binomial distribution

In a bombing attack there is $50\%$ chance that any bomb will hit the target. Two direct hits are required to destroy the target completely. How many bombs must be dropped to give a $99\%$ chance or ...
0
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1answer
18 views

Bivariate normal concave downwards

The Hessian matrix of the bivariate standard normal distribution (where both standard normal distributions are independent) is positive definite. Yet, it concaves downwards. If the Hessian is positive ...
0
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0answers
13 views

Maximum-a-posteriori estimation with Gamma prior and scale-invariant likelihood

Let $\mathbf{X}$ be a vector of parameters with prior distribution $X_i \sim \text{Gamma}(\alpha, \beta)$ i.i.d. for $i = 1, \ldots, n$. Let us denote this prior by $p(\mathbf{X})$. We get to observe ...
0
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0answers
20 views

expectation of discrete uniform distribution (general case)

You may be thinking - can't he find this answer somewhere else? Well I've tried but my textbook is very concise and questions so far have been very problem specific. Well I actually have the answer ...
1
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2answers
26 views

What is the distribution of the square of a sum of correlated Gaussian random variables?

Suppose a random vector $X\in\mathbb{R}^n$ follows a centered multivariate Gaussian distribution with zero mean and covariance $\Sigma$. We know that a linear combination of every elements in $X$ ...
1
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0answers
51 views

Proof Attempt: Non-decreasing continuous CDF is standard uniformly distributed

Proof Attempt: For any random variable $X$ with non-decreasing continuous cdf $F(x)=\Pr(X≤x)$ (note that the inverse function does not necessarily exist due to flat regions in $F$), I wish to prove ...
1
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1answer
20 views

Inferring absolute continuity of summands from absolute continuity of sum

Suppose we have i.i.d. random variables $X_1, X_2$. If $\text{Law}(X_1 + X_2)$ is absolutely continuous with respect to the Lebesgue measure $\lambda$, can we infer that each $X_i$ is absolutely ...
0
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0answers
13 views

How to find (GEV) distribution parameters with optimization?

I'm currently trying to replicate this study with python. http://pages.stern.nyu.edu/~sfiglews/Docs/RND_draft7.pdf The section I'm currently working on is between p.17-20 in the study. The study ...
0
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0answers
12 views

Comparing two samples with weights

Given two sets of samples and their respective weights, what is a sensible way sensible way of comparing these two, i.e., if they are representing the same distribution? Some things that came into my ...
1
vote
1answer
56 views

Confusion about the average distance traveled on a $1$D random walk

The average absolute distance on a one dimensional random walk is supposed to be $\sqrt{n}$. Where $n$ steps are taken from the origin or $n$ is the time. I don't have an intuitive understanding or ...
0
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0answers
6 views

Find the relation between mean and variance for lognormal distribution giving as input mean and standard deviation of normal distribution

I am working with a Lognormal distribution with mean $m$ and variance $v$. I give as input $\mu$ and $\sigma$ of the relative normal distribution in order to calculate the cumulative Lognormal. Now, ...
1
vote
1answer
26 views

Setting up the liklihood distribution for Bayesian Estimation

Here is the exact problem: Suppose that the random variables $Y_1,\ldots,Y_n$ satisfy $$ Y_i=\beta x_i+\varepsilon_i, \quad i=1,\ldots,n. $$ where $x_1,\ldots,x_n$ are fixed constants, and $\...
2
votes
2answers
63 views

How to calculate variance of W? Find the probability distribution of W?

$W=Y-X$ I have figured out that $E(W)=0.3$ by using this formula $E(X+Y)=E(X)+E(Y)$. I tried using the same formula with $E(X^2)$ and $E(Y^2)$ to find $E(W^2)$. I also tried using $V(X+Y)=V(X)+V(...
3
votes
1answer
18 views

Intuition of the relation between poisson process and order statistics

Lemma: Let $T_n$ be the time of the nth arrival in a Poisson process and $U_k$, $k=1,2....n$ be independent uniform on $(0,1)$. Then the order statistics of $U_1, U_2,....,U_n$ have the same ...
0
votes
1answer
15 views

Find the Expectation of $2\sum\limits_{i=1}^n \frac {(y_i-\alpha)^2-\beta ^2}{(\beta^2+(y_i-\alpha)^2)^2}$ for $y_i$ i.i.d. Cauchy$(\alpha,\beta)$

Find the Expectation of $$2\sum_{i=1}^n \frac {(y_i-\alpha)^2-\beta ^2}{(\beta^2+(y_i-\alpha)^2)^2}$$ given $y_1,y_2...$ iid ~Cauchy$(\alpha,\beta)$ with pdf $(-\infty < y<\infty , \beta>0)$: ...
1
vote
1answer
22 views

The problem of ksdensity plot in Matlab

I want to verify that random numbers generated by exprnd match the exponential distribution. I used Matlab's ksdensity function –...
2
votes
1answer
38 views

Problem involving Central Limit Theorem

The following problems is from Durrett 3.4.9: Suppose $X_i$ are independent and $S_n = X_1 + ... + X_n$. Assume that $$ \begin{split} P(X_m = m) &= P(X_m = -m) = m^{-2}/2, \text{ and }\\ P(...
0
votes
1answer
17 views

Expected Frequency distribution of numbers for a pick 5 from 35 lottery.

There is a Pick 5 from 35 lottery called Mass Cash. It has had 4,073 drawings over its history so far. What would be the expected frequency of a given number being drawn, assuming each number has ...
1
vote
1answer
56 views

1D random walk probability distribution

I am way more physicist than mathematician and this question arises from experimental physics/engineering. The motivation is dealing with small amount of random discrete shifts between measured ...
1
vote
1answer
38 views

$X_i = \mathcal{N}(0, \sigma_i^2)$, with $\sigma_1^2 \geq \sigma_2^2 \geq \dots \geq 0$ and $\sum_i \sigma^2_i = 1$

Let $\{X_k\}$ be independent random variables such that $X_i = \mathcal{N}(0, \sigma_i^2)$, with $\{\sigma_i^2\}$ such that $\sigma_1^2 \geq \sigma_2^2 \geq \dots \geq 0$ and $\sum_i \sigma^2_i = 1$. ...
4
votes
1answer
19 views

Distribution of sums of random variables over finite field

Let $q$ be an odd prime, $X_1, \ldots, X_n$ be i.i.d. random variables over $\mathbb Z_q$, and $0 < p < 1$ be some constant. Let $X_i$ take on the value $0$ with probability $p$, and the ...
-1
votes
0answers
15 views

Expected value of $\min$ condition with normally distributed variable

I want to calculate the value of the term $y=\min \{c, N \}$ where $c$ is a constant number and $N$ is a normally distributed variable. So far I couldnt find anything on the internet and hope that ...
1
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1answer
26 views

Intuition of the Mean wait time in queuing system

In queuing theory, (with a single queue and a single server) , given A is service rate (of customers) and B is arrival rate(of customers) We know that, the average time a customer waits in the ...
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0answers
45 views

How can I generate random samples from following probability density function?

Let $\mathbf{\alpha}=(\alpha_1, \ldots, \alpha_m)$. The posterior density function of $\mathbf{\alpha}$ is given by $$h_0(\mathbf{\alpha}|\mathbf{x})=‎\frac{\prod_{i=1}^{m}\alpha_i^{a_i}}{\left(1+\...
2
votes
2answers
30 views

how far the distribution from the uniform distribution

I have two discrete probability distributions $P$ and $Q$, where $P=(p_1,...,p_n)$ and $Q=(q_1,...,q_n)$, in addition I have uniform distribution $U=(\frac{1}{n},...,\frac{1}{n})$. The question is ...
0
votes
1answer
36 views

Computing this conditional expectation?

Consider two random variables $X_1$,$X_2$ iid from a cdf $G$ on $[0,b]$. Consider also some concave function $f(.)$ with inverse $\phi(.)$ over $[0,b]$ such that $f(x)\leq x$, $\forall x\in [0,b]$. ...
0
votes
1answer
42 views

Find pdf given moments (well first by finding mgf!)

Suppose a continuous random variable has odd moments = zero and even moments as follows $$E[X^{2n}] = \frac{(2n)!}{2^n n!}$$ Then the mgf is, by Maclaurin series expansion where we can switch sum ...
0
votes
1answer
22 views

Distribution of the ratio of exponential r.vs with additive constant

I'm interested in the distribution of $\frac{X}{c+X+Y}$, where $X$ and $Y$ follows the exponential distribution with rate parameter $\lambda$, and $c$ is a constant. I know that without the constant $...
1
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0answers
60 views

Basic query related to conditional expectation

I have two variables $X$ and $Y$. I want to find the following probability $$P(X>a(Y+c),X+Y>d)$$ where $a>0,c>0,d>0$. To find the solution I have done following steps $$E_Y[P(X>a(Y+...
-2
votes
1answer
19 views

conditional distribution of minimum of a constant and a random variable [closed]

Let $S$ be an Exponentially distributed random variable with parameter $\lambda$. We define $T=\min(S,L)$ where $L$ is a fixed constant. In this case, what is the conditional distribution $P(t\mid s)$?...
0
votes
1answer
17 views

Sample space in probability computing

This is a simple example of probability computing. There are $n$ white balls and one black ball in a box. Take the balls one by one out of the box until the black ball appears. Let $X$ denotes the ...
0
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0answers
13 views

Expectation of $Y=X\mathbf 1_{X>t}$ in terms of the CCDF $P(Y>x)$

I have random variable $Y=XU(X-t)$ (here $U(X)$ is the unit step function, $Y$ has non-negative support and depends on other random variable $X$) which has a CCDF $P(Y>t)$. I want to write the ...
2
votes
1answer
14 views

Proof that the sum of two independent exponential random variables is gamma with $\alpha=2$

I'm trying to prove that the sum of two exponential random variables is gamma. This proof is straightforward using the uniqueness of moment generating functions however I'm asked to find the density ...
0
votes
1answer
43 views

Can we use a symmetry argument instead of integration in BASIC probability?

Suppose $H$ is a random variable with pdf $f_H(h)$. Let $X$ and $Y$ be random variables with joint pdf $$f_{X,Y} = f_H(x) f_H(y)$$ Prove $$P(X \ge Y) = 1/2$$ Is it possible to ...
0
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0answers
38 views

Prove that if $X$ and $Y$ are independent, then $h(X)$ and $g(Y)$ are independent in BASIC probability — can we use double integration?

In advanced probability we can just say: \begin{align} & P(h(X) \in A, g(Y) \in B) \\[6pt] = {} & P(X \in h^{-1}(A), Y \in g^{-1}(B)) \\[6pt] = {} & P(X \in h^{-1}(A)) P(Y \in g^{-1}(B)) \...