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

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32 views

Conditional Coin Probability:Will The Decision Change

A decision making problem will be resolved by tossing $2n + 1$ coins. If Head comes in majority one option will be taken, for majority of tails it’ll be the other one. Initially all the coins were ...
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1answer
39 views

Probabability of Joint Distribution

Let the continuous random variables $X$ and $Y$ have the joint probability density function given by $f(x) = 3/2x$ for $0<x<2$, $0<y<1$, $x<2y$. Find Pr $(x<1.5|y>0.5)$. This was ...
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0answers
123 views

How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
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2answers
27 views

Inequality in proof that $ X_n \overset{P} \to X \Rightarrow X_n \overset{d} \to X $

I am looking at the proof of convergence in probability implying convergence in distribution. The proof begins by stating that if $X_n \leq x$ then either $ X \leq x + \epsilon $ or $ |X_n - X| > ...
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1answer
62 views

Chi square distribution

K is chi-square distributed with 33 as degree of freedom. P(x ≤ K ≤ y) = 0,73 and the probability of the above and below [x,y] is equal. I'm supposed to define x and y. I tried solving it like this: ...
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1answer
236 views

What does “taking expectation w.r.t some random variable” mean in this probability calculation?

I am trying to calculate the following probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where, $$A_i \sim \exp(\lambda), \quad S_i \sim ...
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1answer
96 views

Find the distribution of sum of random variables given bivariate distribution.

$\bullet$If $(X1, X2)$ be a bivariate Gaussian random variable with parameters $µ$ and $Σ_{x}$. Find the distribution of $X1 + X2$. Hi all, for this question I'm not sure about the best way ...
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28 views

What is the density of $y|z$ in the following problem

I have three random variables: $x$, $y$, $z$ in $\mathbf{R}$. I know the following about their distributions: $x \sim \text{unif}[-\infty, \infty]$, $y \sim \mathcal{N}(x, \sigma)$, $z \sim ...
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2answers
186 views

Continuous Bivariate Random Variable, Conditional Probability Problem

I am trying to study Bivariate Random Variables. The question is if joint pdf is given by $$ f(x,y) = \begin{cases} 8xy & 0<x<1 \hspace{2mm}\text{ and }\hspace{2mm} 0<y<x \\ 0 ...
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1answer
45 views

Variance of discrete probability distribution

I was wondering how I should calculate the variance of the following discrete probability distribution: $$P(y = 0|X) = w + (1-w)e^{-\mu}$$ $$P(y = j|X) = (1-w)e^{-\mu}\mu^{y}/y! \qquad j=1,2...$$ ...
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1answer
36 views

Find the Conditional Probability

Let the continuous random variables $X$ and $Y$ have the joint probability density function given by $f(x)$ = $kx$ for $0<x<2$, $0<y<1$, $x<2y$. Find $k$. I found the joint ...
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1answer
60 views

Determine Random Variable

We have a random variable $X$. Given the values for $E(X), E(X^2), E(X^3), ...$, is it possible to determine the distribution of the random variable X? PS: Here $E(X)$ is the expected value of the ...
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1answer
85 views

What is the joint CDF of f(x,y)=2(x+y) 0<=x<=y<=1

I am trying to find the joint CDF of $f(x,y)=2(x+y) : 0\leq x\leq y\leq 1$. There are five different answers for the CDF depending on the restrictions of $x$ and $y$ that you use. I found the CDF ...
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2answers
42 views

Given the joint pdf of random variables $X_1$ and $X_2$, I'm trying to find the pdf of $W=w_1X_1 + w_2X_2$

I'm given $f_{X_1,X_2}(x_1,x_2)=e^{(-x_1-x_2)}$ where $x_1>0$ and $x_2>0$. Also, $w_1$ and $w_2$ are both constants greater than zero. I'm trying to first determine the cdf of $W$ for which I ...
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1answer
158 views

How to derive this pdf?

I understand how to find the pdf for the sum of $N$ exponentially distributed random variables, but how do I find the pdf when $N$ is also an independent random variable. Here is the problem: Let ...
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1answer
180 views

Probability Density Function of Scaled Gamma Random Variable

Assume we have a Gamma Random Variable $X$ with the following pdf $$ \frac{m^mx^{m-1}}{\Gamma(m)}\text{exp}(-mx)$$ If I am asked to find the distribution of the following $$Y= aX$$ where a is ...
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19 views

Inverse weighted mean there the lightest values are the most important used with uncertainty of values

I have a few values which are results of measurement. Every of these values has certain uncertainty, given as absolute portion of the value (eg. $y_i = 16\pm 0.5$, uncertainty $u_i = 0.5$, $x_i = ...
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1answer
142 views

How to calculate the probability of two events happening within a certain time period using exponential distribution

I know how to calculate the probability of one event taking place within a set time period with exponential distribution but I'm having difficulty figuring out how to calculate what would happen if ...
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1answer
54 views

“With high probability” statement from CLT

Suppose $X_1,X_2,..,X_n$ i.i.d. with mean $\mu$ and variance $\sigma^2$, so that \begin{equation} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\longrightarrow ...
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1answer
45 views

Calculation of PDF of derived multivariate random variables?

Let we have $X$, $N$ dimensional vector of independent random variables. If we multiply this vector by some matrix $V$ with size $r\times N$, with property $V*V'=I$, where $I$ is identity matrix, and ...
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1answer
32 views

Sum of three probability density functions.

If $A_1,A_2$ and $A_3$ are independent and have pdf $f(t)=\lambda e^{-\lambda t}$ when $t \geq 0$ and $0$ when $t<0$. What is the pdf of the sum $A_1+A_2+A_3$. So the first step is the pdf of sum ...
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1answer
48 views

Binomial cumulative probability

Here is the question I need help on : Let $X$ be a binomial random variable with p = 0.5 and n = 100. Give $P(X \geq 60)$ rounded to two decimal places without using a calculator (by using ...
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1answer
37 views

Exponential distribution lightbulb

The time it takes for a lightbulb to burn out is exponentially distributed with mean $u$ which is a random variable. Asssume that $u$ is distributed with density $$f(x)=\frac{8}{x^3}$$ for $x \in ...
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1answer
28 views

Calculating the mass function of maximum of a sum

Find an expression for the mass function of $N(t)$ in a renewal process whose interarrival times $X_i$ are a) poisson distributed with paramter $\lambda$ and b) gamma distributed $\Gamma(\lambda,b)$. ...
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2answers
53 views

Sum of probability density functions

If $A_1$ is random variable exponentially distributed with pdf $f(t)=\lambda e^{-\lambda t}$ when $t \geq0$ and $0$ when $t< 0$. Let $A_2$ be random variable that is independent of $A_1$ but it ...
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2answers
518 views

Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I ...
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4answers
42 views

Probability distribution (Binomial distribution)

Let's say we have a coffee shop that only sell coffee and tea. Based on the past selling records, 40% of the customers order tea.(customer will either order tea or coffee - not both) Now, for the ...
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2answers
147 views

What is the expected value of the payment if T has exponential distribution with mean 5?

The initial value of an appliance is $700, and its future value is given by: \begin{align}v(t)=100(2^{3-t}-1),&&0\leq t\leq 3.\end{align} If the appliance fails in the first 3 years, the ...
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1answer
54 views

Mgf of double exponential RV

In class the other day we were talking about a double exponential RV $X$ with a pdf $f(x)=\frac{1}{2}e^{-|x|}$ for $-\infty<x<\infty$. The professor noted that the mgf was $M(t)=\frac{1}{1-t^2}$ ...
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1answer
37 views

Calculating Expectation from CDF

The CDF is defined as follows: $$ F(x) = \begin{cases} 0,\qquad x \lt 0 \\[3ex] \frac{x^2}{18}+\frac{x}{6}, \quad 0 \le x \lt 3\\[2ex] 1,\qquad x \ge 3 \end{cases} $$ And i have to calculate the ...
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1answer
154 views

Normal distribution - how to solve P(-b<X<b)=0.95

$X\sim N(2,3^2)$ How do you find $b$ where $P(-b<X<b)=0.95$ other than trial and error? You can't directly transform to $z$ because if you find an appropriate $z$, transforming back will give ...
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1answer
92 views

Joint distribution $(X_1,X_{(n)})$ order statistics

Let $X_1, \ldots, X_n$ a random sample of a Uniform(0,1), I want to show which the joint distribution of $(X_1,X_{(n)})$ is. I do the following: $$ P(X_1\leq x, X_{(n)}\leq y)=P(X_1\leq x, X_1\leq y, ...
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0answers
287 views

Maximum Likelihood Estimation with Laplace Distribution

I want to estimate the parameters $a$ and $b$ of the model $y_i = ax_i + b + \varepsilon_i, i=1,...,n $ via Maximum Likelihood. The $\varepsilon_i$ are assumed to be Laplace-distributed with density ...
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1answer
40 views

$\delta=0$ in Lyapunov's condition of CLT

In my textbook, Lyapunov's condition is shown to imply Lindeberg's condition by: ...
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1answer
155 views

Random distribution: Parabola distribution (non-linear distribution)

Thanks to a very helpful answer, I recently successfully implemented a linear probably distribution (for my open source constraint solver) to select an element out of ...
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1answer
35 views

Exponential Distribution - Generate a fairly hefty data set X that is approximately exponentially distributed with μ = 0.7.

I encountered a question in one of the books, which asked to generate a fairly hefty data set X that is approximately exponentially distributed with μ = 0.7. I can do it for approximately normally ...
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1answer
46 views

Is the following possible?

I am confused about a simple question: Is $X\sim N(\frac{1}{2},\frac{1}{4})$ possible? My answer is no, because of the $Var[X] = E[X^2]-m_x^2$ If it is true, then $E[X^2]=0.5$, which is impossible. ...
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1answer
31 views

I am trying to figure out where a CDF F(x,y) is a bivariate CDF

I am trying to figure out where a CDF $F(x,y)$ is a bivariate CDF. I know the properties that satisfy it is a bivariate CDF, I just do not know how to show they do not satisfy them (I know it is not a ...
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2answers
80 views

Show the diffusion equation is a normalised distribution.

The diffusion equation is defined to be $$P(x,t) = \dfrac{1}{\sqrt{4D\pi t}} \exp \left(-\dfrac{x^2}{4Dt}\right),$$ where $D$ is a physical constant. Show that the reaction diffusion equation is a ...
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1answer
280 views

poisson gamma mixture

Let $N$ be Poisson distributed with parameter $10B$, where $B \sim \Gamma(3,1)$ (i.e. $f(b)=\frac{b^2 e^{-b}}{2}$). Find the p.m.f of $N$. How should I manipulate $10B$ in the integration? What is ...
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95 views

Uniform Probability Distribution 1

A manager of a department store reports that the time of a customer on the second floor must wait for the elevator has a uniform distribution ranging from 2 to 4 minutes. If it takes the elevator 30 ...
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117 views

Two non-negative integers are chosen at random, then the probability that the sum of their squares is divisible by 5 is?

I saw answer to a similar question but I want mathematical answer not just a view-able thingy. what I did was same as in that question but that got too long, I tried it for first 10 numbers, but the ...
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189 views

About the total number of twin primes in the vicinity of twin primes

Just for curiosity's sake, I did a test regarding twin primes, and I have doubts about the meaning of the results. Test: calculation of ${\pi_2}$(n) and the twin primes density in the vicinity of ...
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1answer
44 views

Determine “winner” in exponential contest

If I have one light bulb that could be one of $2$ kinds, ($A$ and $B$ are the lifetimes of first and second type: $A\sim \exp(1)$ and $B\sim \exp(3)$), and each time a bulb dies, another bulb replaces ...
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1answer
26 views

Probability function and random variables

Given a Bernoulli r.v., W, which is derived from r.v. T(Poisson) (a)if T=0 then W=1 and b) if T>0 then W=0). One has to show that the sample mean (the proportion of 0s in the sample), is an ...
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2answers
351 views

Calculating variance of marginal distribution

Suppose you have a random variable $Y \sim Po(\mu)$ where $\mu$ is given by $\mu \sim Ga(\alpha, \eta)$. Correct me if I'm wrong, but I think it is clear that the Poisson distribution is a conditional ...
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3answers
59 views

What is the joint distribution of random variables Y and XY?

Random variables $X$ and $Y$ are independent. $X$ takes on values of $-1$ and $1$ with $\frac{1}{2}$ probability each, $Y$ takes on values of $-1,0,1$ with $\frac{1}{3}$ probability each. What is the ...
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1answer
65 views

Sum of two random variables. Limits of integration

Let $X$ have pdf $$f_X(x)=e^{-x} \qquad \text{ for } x \ge 0$$ and $Y$ have $$f_Y(y)=1 \qquad \text{ for } 0\le y \leq 1$$ $X, Y$ both independent. What is the pdf for $Z=X+Y$? Using convolution ...
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1answer
325 views

Markov's Inequality for Negative Binomial distribution

Given that $Y$ follows Negative Binomial distribution (counts y successes before $k$th failure), using Markov's inequality show that for any $q \in[p,1]$, there exists constant $C$, such that ...
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1answer
104 views

Tricky Negative Binomial example

Let $Y$ count the number of widgets succesfully produced before $r$th failure. We are told that machine shuts down when $30$th failure has occured, that is $r=30$. Then probability of producing $y$ ...