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

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0
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1answer
110 views

Probability of watching TV or Cartoons

so I am trying to solve this problem: Let X and Y equal the respective numbers of hours a randomly selected child watches movies or cartoons on TV during a certain month. From experience, it is know ...
0
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1answer
170 views

CDF of a density function with absolute value.

If X is a random variable with the density function $f(x)=\frac{e^{-|x|}}{2}$, what is the CDF of X? My first inclination is to take $\int_0^\infty \mathrm{e}^{x}/2\,\mathrm{d}x$ and ...
1
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2answers
111 views

Prove variance in Uniform distribution (continuous)

I read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?
2
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0answers
59 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
1
vote
1answer
22 views

Exchangeability with random effects?

Consider a $N\times N$ random matrix $$ \epsilon:= \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & \dots &\epsilon_{1N} \\ \epsilon_{21} & \epsilon_{22} & \dots & ...
4
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2answers
77 views

Understanding the Beta-function

I always forget whether the beta function, B$(\alpha, \beta)$, is defined as $\Gamma(\alpha+\beta)/\Gamma(\alpha)\Gamma(\beta)$ or $\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$. Is there an ...
0
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1answer
51 views

Calculating Variance and Standard Deviation with probability distribution

The age [in years] $X$ of sewing machines to be reconditioned is a random variable with the following probability distribution: $f(x)=(1/972)x(18-x)$ for $0<x<18,$ and $f(x)=0,$ elsewhere. The ...
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0answers
29 views

Finding Variance and Sigma

Let $X$ be a random variable with $E(X)=3$ and $E[X(X-1)]=22$ $(i)$ Find $Var(X)$ I am not sure about this one. This is what I did though: $Var(X)=E[X^2]-E[X]^2$ $E[X(X-1)]=22$ ...
0
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1answer
211 views

the random heights of north american women

The heights of North American women are normally distributed with a mean of 64 inches and a standard deviation of 2 inches. A random sample of four women is selected. What is the probability that the ...
1
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1answer
84 views

determining the size of a test bank given acceptable number of repeats

I have a question for a challenge that I'm trying to create - having some trouble quantifying the size of the challenge's test bank. 20 people are taking a challenge of 9 questions the test bank (n) ...
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3answers
158 views

Conditional Expectation of rolling two dice.

I have to find $E(X\mid Y)(y)$ where $X$ is the value of the first roll and $Y$ is the sum of the two dice. I know that $$E(X|Y)(y) = \sum_x{xP(X\mid Y)}=\frac{\sum_xxP(X=x, Y=y)}{P(Y=y)},$$ but ...
0
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1answer
407 views

How do you transform Gamma to Chi-squared distribution

Here is the question not sure how to turn a Gamma into a Chi-Squared: Suppose $X_1....X_n$ is a sample from the distribution Gamma($\alpha=3,\ \lambda=\theta$) with unknown $\theta > 0$. We wish ...
0
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1answer
54 views

derive the mean and variance of $\bar X$ using means of sums rules

I can't find anywhere what the means of sums rules are so i'm confused with this question The random variables $X_1......X_5$ are jointly multivariate normal. Their expectations are $E(x)= \mu_i$ and ...
0
votes
1answer
67 views

Calculate $E[XY]$ of dependent variables

I'm having a little trouble whit a probabilistic exercise. The problem says this: There's a vase whit 10 marbles, 4 black and 6 white. Now I extract 2 of them without reposition. Being $X,Y$ random ...
2
votes
1answer
126 views

Gumbel distribution

Let $(X_i)_{i \geq 1}$ be a sequence of i.i.d. normal $\mathcal{N}(0,1)$ random variables. Let $M_n = \max_{i=1,\ldots,n} X_i$. Show that $$P[\sqrt{2 \log n} M_n - 2 \log n \leq u ] \rightarrow ...
1
vote
1answer
110 views

Probability distribution of the product of random numbers

For applied mathematics to evolutionary biology I am often faced to have to describe a probability distribution function (PDF) which results from the product of a function in which a parameter is ...
0
votes
1answer
59 views

Gaussian Approximation of an intractable distribution

I am currently encountering this problem: I have an intractable distribution and I want to minimize the KL divergence of this distribution and a multivariate gaussian distribution. So we just need ...
0
votes
1answer
31 views

Variance of a linear combinations

I was given the problem above. I am confused on how to find the variance of the linear combinations. A for example would have a mean of 22 correct? Can someone ...
1
vote
1answer
54 views

Determine a distribution with no parameters?

I'm confused by this question and I was hoping for some guidance some one to point me in the right direction Let $X_1.........X_n$ be a random sample from a population with mean $\mu$, that is ...
0
votes
1answer
35 views

Limit in probability of $(\bar X)^2$

Hello can some one point me in the right direction with this question? Let $ F(x) = \begin{cases} 1- x^{-4} & x>1 \\ 0 & elsewhere \\ \end{cases} $ be the CDF of a random ...
2
votes
2answers
343 views

Professor has 4 umbrellas, Markov chain and Probability

OK this problem is making me tear my hair out. I need someone to walk me through this in baby-steps method like 1 + 1 = 2. I am trying to figure out what I don't understand. I know this is going to be ...
2
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3answers
733 views

Random variable with infinite expectation but finite conditional expectation

I've been very stuck on a question from Probability and Random Processes by Grimmett and Stirzaker for ages - so stuck that I flicked to the back to have a look at the answers. But, I can't seem to ...
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0answers
132 views

Determine the Maximum Likelihood Estimator of $\Theta$ and Consistency

Alright so I have been working with this one for a while and i'm not totally sure where to take it Let $X_1,.....X_n$ be a sample from a distribution with CDF $F(x;\theta)= 1-C/((x-\theta)+3)^4 for$ ...
1
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0answers
56 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
0
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1answer
31 views

Which probability distribution is this?

Suppose we draw a number $x$ uniformly distributed on $(0,1)$, what is then the following distribution. Furthermore, calculate $F(y)$ and $f(y)$. $$y = \dfrac{x}{1-x}$$ This is a question I came ...
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1answer
73 views

Finding the density function from joint density function

I'm reading the conditional distributions section of Probability and Random Processes by Grimmett and Stirzaker and I've come across a brief exercise I can't seem to figure out. We're given earlier ...
0
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0answers
703 views

Expected length of a stick broken into two pieces — explaining the length function and PDF

From the question on this page : We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece In one of the answers, the following ...
3
votes
1answer
45 views

Limit of independent sequence of Normal Distribution is Chi Distribution

I am looking for some help to solve the following central limit theory question below Let $X_1, X_2, \ldots$ be i.i.d. with $X_i ≥ 0$, $EX_i = 1$, and $\operatorname{var} (X_i) = \sigma^2 \in (0, ...
0
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1answer
218 views

The Delta Method asymptotic distribution

I've managed to get through a) and b) but now I'm stumped with this delta method and cramer theorem stuff: let X be Gamma random variable with parameter $\alpha$=4 and $\lambda$=$\theta$: a)Find the ...
2
votes
1answer
346 views

Exponential Distribution with changing (time-varying) rate parameter

In short, how would one sample an Exponential Distribution with an exponentially-increasing rate parameter, e.g., $\lambda=e^t$? What distribution does such a random variable follow? Background: The ...
1
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1answer
125 views

Distribution of a sample generated from an AR(2) model

Consider the autoregressive model of order 2 $$X_{t}=\varphi_1X_{t-1}+\varphi_2X_{t-2}+\varepsilon_t,$$ where $\varepsilon_t$ are zero-mean normally distributed random variables with $\sigma^2$ ...
1
vote
1answer
49 views

Addition corresponds to convolution and subtraction?

We know that if two random variables have proper densities, than the density of the sum of them is given by the convolution. But what can we say about the difference of two random variables? $X-Y$ ...
0
votes
1answer
87 views

Probability Problem; which distribution to use and how to find the “mean”?

I've been working on my homework for a while, but I got stuck with this question: A  trading  company  has  nine  computers  that  are  used  to  trade  on  the  New  York  Stock Exchange.  The  ...
0
votes
2answers
34 views

Maximum Likelihood from observed values

Give IID Data Samples $X_n = $ {$x_1, x_2, ..., x_n$} generated from a uniform distribution $U(x|0,\theta)$. $p(x|\theta) = U(x|0,θ) = ${$ \frac{1}{\theta}$ for $0 \leq x \leq θ$ and $0$ otherwise}. ...
0
votes
1answer
2k views

How do I find if the probability of the sample proportion is greater than something?

I have this problem and I have no clue how to solve it. In 2012, 31% of the adult population in the US had earned a bachelor’s degree or higher. One hundred people are randomly sampled from the ...
0
votes
1answer
45 views

Given the distribution of a random variable $R$, who do you get a uniform random variable $U$?

Let us say you have a random variable $R$. How would one generate a uniform random variable $U$, with the maximum possible entropy (or infinite entropy, if $R$ has such)? (For simplicity, you may ...
0
votes
1answer
22 views

Probability densitiy of $\cos x$

If $x$ is of Uniform distribution between $0$ and $2\pi$, what is the probability of $\cos(x)$ having a value between $-0.5$ and $0.5$? I tried transformation, but I somehow get $1/6$ but it seems to ...
0
votes
1answer
20 views

Generate samples from other samples

Given a family of continuous random samples $(x_i)_{i \in I}$ that approximate some unknown probability distribution. How can I generate more samples that fit to the same unknown distibution? ...
1
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0answers
36 views

Special distribution with high probability of being zero and fixed second and fourth moment

I have the following problem: Im looking for a probability distribution (i.e. a random variable with this distribution) that fullfills the following properties: $\mathbb{P}(X=0)=\delta$ where ...
0
votes
0answers
273 views

Find the best predictor and the best linear predictor of $Y^2$ given $X$. Suppose $(X, Y ) \sim N(0, 0, 1, 1, p ).$

Once more, there's another question that I'm clueless on how to start. I should have dropped this course earlier. Suppose $(X, Y ) \sim BN(0, 0, 1, 1, p )$, meaning that $X$ and $Y$ are bivariate ...
1
vote
1answer
29 views

Calculate a probability involving drawings from bivariate normal variables with Xi and Yi i.i.d

There's a question which has been troubling me along with my earlier post. To be honest, I'm not entirely sure on how to proceed. All I know is that if X~N(mu,sigma^2) then P(X < A) = P(Z< ...
1
vote
1answer
175 views

Expectation of (1/x)-1 possible transformation involved??

I'm a bit confused with the first steps in this problem: $F(x)=x^4$ for $0<x<1$ a) Find $E[(1/X)-1]$ b) Let $Y=(1/X)-1$. Find the support of $Y$, its pdf and CDF. Name its ...
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1answer
71 views

Product of exponential distributions

Suppose $X_1$ is $\mathrm{Exp}(\lambda_1)$ and $X_2$ is $\mathrm{Exp}(\lambda_2)$. $X_1$ and $X_2$ are independent. Let $Y = \min (X_1, X_2)$ and $Z = \max (X_1, X_2)$ and $W = ZY$ . Compute the ...
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0answers
35 views

Get distibution depending on three random variables

Given the function $\text{result} = \frac{a*b}{1-c}$ with 3 independent equally distributed random variables $a$, $b$, and $c$, how do I derive the distribution of $\text{result}$? How can I get the ...
0
votes
1answer
146 views

Find $\operatorname E(X\mid Y)$ given that $X=U+V$ and $Y=UV$ when $U$ and $V$ are independent with exponential distribution.

I currently have a problem with a problem set I'm working. Suppose $U$ and $V$ are independent with exponential distribution with parameter $A$ ($T$ is exponentially distributed with parameter if ...
0
votes
1answer
68 views

Moment generating functions and expectations

This may be a very stupid question but I just want some clarification. Take for example a geometrically distributed random variable $Y$ with PMF $p_Y(y) = p(1-p)^{y-1}$ and MGF $\mathbb{E}[e^{tY}] = ...
3
votes
1answer
21 views

prove tail probabilities equation

Let's say $p(z) = E[z^X]$, which is a probability generating function of a random variable X. Could we prove following equation? $\sum_{k \geq 0} Pr[X \geq k] z^k = \frac{1-zp(z)}{1-z}$ in which ...
0
votes
2answers
57 views

Expectation of a function of two different transformations of a random variable

I have a random variable $X$ where the support of $X$ is non-negative and I want to compute the expectation of $aX\log(bX)$ where $a,b>0$ If I can chose how $X$ is distributed, is there a ...
1
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1answer
93 views

Joint Density of $\sin(2\pi U)$ and $\cos(2\pi U)$,$ U$ uniform $(0,1)$

I need help with finding the joint density of $X$ and $Y$ where $$X=\sin(2\pi U),\quad Y=\cos(2\pi U)$$ where $U$ is uniformly distributed on (0,1).
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6answers
537 views

How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...