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

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1answer
29 views

Given $f(x,y) = 1$, $0<x,y<1$, let $U = X+Y$. Find $f_U(u)$.

Would anyone be able to explain what they did in the second line, especially how they got $0<u<1$, and $1<u<2$
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1answer
1k views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, $...
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1answer
35 views

Discrete Uniform Probability: isn't my textbook just wrong?

My textbook is showing me examples of discrete probability distributions, one of them is in the picture: I learned in Calculus that the summation of the series $1/n$ where $n\to \infty$ is ...
0
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1answer
27 views

What type of distribution is this?

$$p(x) = \frac{1 }{\theta} . \frac{(ln\theta)^x}{x!} $$ I have no idea what type of distribution is this? And what will be its first and second moment?
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1answer
1k views

Proof for Standard Deviation Formula for a Binomial Distribution

I understand the concept of standard deviation as the square root of the square of the mean of each sample value - the mean of the sample values. Here is the mathematical representation (I've solved ...
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1answer
3k views

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed?

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed? Any explanation would be very appreciated.
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0answers
35 views

Let $X$ be a random variable with cumulative distribution function $F_X$, find $F_Y$ for $Y = \sin(X)$ and $Y = \cos(X)$

Let $X$ be a random variable with continuous cumulative distribution function $F_X$, find $F_Y$ for $Y = \sin(X)$ and $Y = \cos(X)$ My approach: I would say that $F_Y(y) = \mathbb{P}(Y \leq y) = 1$...
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1answer
19 views

Most likely value of negative binomial random variable

If $X$ is a negative binomial random variable let's say with $p =0.2$ and $r = 4$ then how can we calculate most likely value of $X$? I thought it is expected value but that is $20$ and I guess most ...
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0answers
27 views

'Finding' a normally distributed random variable

Let a random variable $Z$ have a standard normal distribution. Suppose that we start at $0$. We 'walk' right, along the number line, till we reach $a$. We then turn around, walk back, past $0$, till ...
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1answer
34 views

How to find value of RV for this condtion [closed]

The probability density function of the time to failure of an electronic component in a copier(in hours) is $$f(x)=\exp(-x/1000)/1000$$ for $x>0$ and $f(x)=0$ for $x\leq 0$. How determine the ...
2
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1answer
69 views

Limit of a multiple integral [closed]

$$\displaystyle\lim\limits_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\left(\frac{\pi}{2n}(x_1+x_2+...x_n)\right)dx_1 dx_2...dx_n$$ I don't know how to begin.
0
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1answer
38 views

Loss probability and VaR

I would like to estimate Value-at-Risk analytically and through delta-gamma aproximation. I don't know if my idea is ok, but i would like to build a portfolio of European option. Suppose that in this ...
1
vote
1answer
25 views

Finding the distribution of a random vector in a conditional probability problem [closed]

Players A and B are playing a game of drawing coins from two boxes without returning/replacing them. Box1 has three coins with values 0, 1 and 2 and Box2 has two coins with values 1 and 2. In the game,...
0
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1answer
34 views

generating distribution density function for a system of events with exponential distribution

We have a system in which events happen after each other. Events are i.i.d. An event, shown by random variable $X$, follows exponential distribution with $E(X)=\frac{1}{\lambda}$. We suppose the ...
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2answers
25 views

Conditional Survival Function

Let $X$ and $Y$ be random variables with joint distribution $F(x,y)$. Let $F(y|x)=P(Y\leq y|X=x)$ denote the conditional distribution of $Y$ given $X=x$. Let $S(y|x)=P(Y>y|X=x)$ the conditional ...
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0answers
15 views

The distribution described by the inverse of the sum of two power law distributions

I'm trying to fit real-world data, and have been trying a whole lot of different distributions, some of which I've been making up on the fly. One distribution that fits the data very well is described ...
1
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2answers
25 views

Show that $Z^2 \sim Gamma(\frac{1}{2},\frac{1}{2})$ where $Z \sim N(0,1)$ using change of variables method

So I found a couple of solutions that use integration but I would like to solve this using random variable transformations. I found a similar solution but was hoping to get some more clarity. Here is ...
0
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1answer
28 views

Discrete probability distribution where (max - min) $\not=$ average [closed]

Let $X$ be a discrete random variable with $N$ possible values and some distribution. What are some well-known distributions for $X$ such that $\mathbb{P}(x_{max}) + \mathbb{P}(x_{min}) \not= \...
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0answers
11 views

Help with finding a particular joint distribution of a Bayesian Network

Consider a Bayesian Network defined by the following matrix: $$\left[\begin{array}{ccccccc} 0&1&1&0&0&0&0 \\ 0&0&0&1&1&0&...
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1answer
27 views

Problem with the density of the compound distribution

My problem is to calculate $E[\max(S-5000, 0)]$ where $$S = \sum_{i=1}^{N} X_i,$$ $N$ is a random variable with geometric distribution, parametrized as follows: $$P(N=n) = \frac{\beta^n}{(1+\beta)^{...
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1answer
16 views

what is the relation between variance and mean for two Laplace distribution variables? [closed]

if X and Y are two Laplace distribution variables. assume that $sx^2$ and mx are the variance and mean for variable X. And $sy^2$ and my are the variance and mean for variable Y. What is the relation ...
2
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3answers
45 views

Expected value of a geometric distribution with first step analysis.

I am trying to understand the "story proof" found in this lecture. I am a bit confused as how the expected value of a random variable differs from the the random variable itself when considering ...
1
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1answer
31 views

Is this a Multinomial distribution?

I find it hard to notice when do I have a Multinomial distribution and if its possible to "transform" problems into a Multinomial distribution problems. For example I have the following exercise: ...
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0answers
19 views

Random variable with characteristic function cosine

So I am searching for a Random Variable $X$, such that $\varphi_X(t)=\cos(t)$. I know how to choose $X$ such that $\varphi_X(t)=e^{it}$ and $\varphi_X(t)=e^{-it}$. Does this help me? How can I put ...
0
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1answer
21 views

CDF probability problem [closed]

For random variable $X$ and $Y$ defined on the same sample space, let $U = \min\{X,Y\}$ and $V = \max\{X,Y\}$. a) Determine the CDF of $V$ in terms of the joint CDF of $X$ and $Y$ b) ...
1
vote
3answers
1k views

Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...
0
votes
2answers
38 views

Expected value of $Y=\prod_{j=1}^{N} X_{j}$, where $N\sim\operatorname{Poisson}(\lambda)$.

Let $$Y=\prod_{j=1}^{N} X_{j},$$ $X_{j}$, for $j=1,2, ...$, are identically and independently distributed with mean $\lambda$ and variance $\sigma^{2}$ and $N\sim \operatorname{Poisson}(\lambda).$ ...
0
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0answers
39 views

Integral of non-Gaussian distribution, random walk?

I would like to evaluate $$ F = \frac{\mathbb{E} \left\{\left(\int_0^T x^3(t) dt \right)^2\right\}}{\mathbb{E} \left\{\left(\int_0^T x(t) dt \right)^2 \right\} } \approx \frac{\mathbb{E} \left\{\left(...
0
votes
2answers
36 views

Expected value of hyper geometric distribution

Question: Say, I have to calculate the expected value of the number of aces from a deck. I pick cards without replacement. Thus, the distribution of the number of the cards is hypergeometric. ...
1
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2answers
43 views

Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$

Let IG denote Inverse-Gamma distribution Inverse-Gamma. If $X\sim IG(\alpha,1)$ and $Y\sim IG(\beta,1)$. Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$ I tried with jacobian transformation ...
1
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2answers
33 views

Finding the probably density function of $Z=\sqrt{X^2+Y^2}$ where Y~N(0,1) and X~N(0,1)

X and Y are normal random variables that are independant. Finding the probably density function of $Z=\sqrt{X^2+Y^2}$ where Y~N(0,1) and X~N(0,1). Attempt: Let $z \in R$. If $z \lt 0$ then $P(Z\le ...
2
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1answer
26 views

Joint distribution weird result

We have 3x3 board Each board cell has $0.5$ chance to be white (there is no dependency between different cell colors) Let X = number of white rows (a row with only white cells on it) Let ...
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2answers
51 views

CDF of sum of N exponentially distributed random variables with condition

I have $Y=X_1u(X_1-x_{th})+X_2u(X_2-x_{th})+\cdots+X_Nu(X_N-x_{th})$, with all the $X_i\sim\lambda e^{-\lambda}$, $u(t)$ is the unit step function and $x_{th}$ being the threshold which means that any ...
2
votes
1answer
188 views

Series of continuous random variables is continuous

We work on the usual $(\Omega,\mathscr{F},P)$. Suppose $X_i$ are independent random variables. Say the distribution of $X_i$ is $F_i$. Under what circumstances can I guarantee that $\sum_{i=1}^\infty ...
1
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2answers
28 views

Right limit of integral

Following is the joint PDF of RV $X,Y$ and $Z$ $$f(x, y, z) =\begin{cases} kxy^2z;& 0 < x,y < 1, 0 < z < 2,\\ 0,& \text{elsewhere}. \end{cases}.$$ To find value of $k$ I tried ...
0
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1answer
23 views

What is the closest apporoximation for pdf of log-normal distribution?

I am unable to compute a complex integral which uses the pdf of log-normal distribution. Hence, I want to replace the pdf of log-normal distribution with an alternate function(s) (piece-wise ...
1
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1answer
29 views

Can I run a regression when both independent and dependent variables are all dichotomous?

I have conducted a survey where all my questions are asked in a dichotomous manner (Yes/No). Eg IV:"Are you a smoker?", "Are you obese", "Is your gender male/Female" etc. DV: "Have you ever had a ...
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0answers
18 views

Contradiction in the CDF derivation from two different strategies

I have a sum $Y=X_1u(X_1-t)+\cdots X_Nu(X_N-t)$ where all $X_i's$ are i.i.d with exponential distribution with parameter $1$ and $u(x)$ is the unit step function. As can be seen from the expression of ...
0
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1answer
16 views

Solving for a Conjugate Prior in search of MAP estimator

I am trying to prove that if a given random variable $X \sim Exp(\lambda)$ and $\lambda \sim Gamma(\alpha,\beta)$ hen $\lambda | X \sim Gamma(\alpha^{*},\beta^{*})$ for some parameters $\alpha^{*}$ ...
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0answers
44 views

Three random variables with exponential distributions

Having $X$, $Y$ and $Z$ as three independent identical random variables all having exponential distribution $E(X)=E(Y)=E(Z)=\frac{1}{\lambda}$, What is the answer of the following probability: $P(X+Y&...
1
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1answer
34 views

Why aren't CDFs left-continuous?

Let $F$ be a cumulative density function on $\mathbb{R}$. From an argument in a textbook, it is shown that $F$ must be right-continuous: Let $x$ be a real number and let $y_1$, $y_2$, $\ldots$ be ...
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1answer
60 views

Let $X$ be a random variable with mean $0$ and finite variance $\sigma^2$. By applying Markov’s inequality show that

I am looking for confirmation that I am working in the correct direction as well as pointers for points where I have gone astray. Here is the problem. (a) Let $X$ be a random variable with mean $0$ ...
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0answers
28 views

Why does the following equality hold in proving Meyer's inequality?

I have a question in proving Meyer's inequality. The proof I read is taken from the book "Malliavin Calculus and related topics" by Nualart. I just have one equality which I am not sure, I will ...
0
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1answer
29 views

Let $X$ have a Poisson distribution with parameter $\lambda$.

Let $X$ have a Poisson distribution with parameter $\lambda$. (a) Show that the moment-generating function of $$Y = \dfrac{(X − \lambda)}{\sqrt{\lambda}}$$ is given by $$M_Y(t)=exp(\lambda e^{\frac{t}...
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1answer
32 views

$P(T ≤ 5 | T ≥ 2)$ from CDF [closed]

If for discrete random variable T the CDF is defined as $$F(t) = \begin{cases} 0, & \text{t<1}\\ 1/4, & \text{1≤t<3}\\ 1/2, & \text{3≤t<5}\\ 3/4, & \text{5≤t<7}\\ 1, &...
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2answers
28 views

Chebyshev's inequality to find probability of interval

Here is how I solved the problem: $$ X\sim N(\mu=.13, \sigma^2=.005^2)\\ .12\le x\le .14 \\ \mu-2\sigma\le x \le \mu+2\sigma\\ $$ Using Tchebychev's inequality, I get $$ P(|x-\mu|\le 2\sigma)=1-\frac ...
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4answers
41 views

Convolution: Give a proof that $f_T(t)=\int_{-\infty}^{\infty}f_X(x)f_Y(t-x)dx$ where $f_T(t)$ is the PDF of random variable T

Here is the question: Let $X$ and $Y$ be independent, continuous r.v.s with PDFs $f_X$ and $f_Y$ respectively, and let $T=X+Y$. Find the join PDF of $T$ and $X$, and use this to give a proof that $...
0
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1answer
14 views

How the value of denominator calculated here?

I found this example in a book and it has to find probability distribution as stated below: If a car agency sells 50% of its inventory of a certain foreign car equipped with side airbags, find a ...
1
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1answer
26 views

Joint probability distribution.

I am trying to calculate P(Y|Z) given the following distribution $\ P(X,Y,Z) = P(X)P(Z)P(Y|X,Z)$ Now, initially I did the following calculation. $$P(Y|Z)=\sum_{x}^{}P(X,Y|Z)=\sum_{x}^{}P(X,Y,Z)/P(Z)=\...
0
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1answer
35 views

What is the probability of getting exactly one two and one three in a 5 card draw?

In a 52 cards deck, what is the probability of getting exactly one 2 and one 3 if 5 cards are drawn. I'm wondering what is the difference between doing it the following two ways. Intuitively I would ...