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

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1
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2answers
7 views

Given E(X) and Var(X) find the Expectation of $E[x-2(X-1)^2]$

Let X be a r.v. with $E(X) = 5$ and $Var(X) = 30$. Find $E[X-2(X-1)^2]$. I'm not sure as to how to approach this problem, any tips on how to approach it would be appreciated!
0
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0answers
7 views

limiting distribution of a function of joint normals

Let $Z_n=(X_{1,n},X_{2,n})\sim N(\mu,\Sigma_n)$ where $\mu=(0,0)'$ and $$\Sigma_n=\begin{bmatrix}a^2+\frac1n & ab \\ab & b^2+\frac1n\end{bmatrix}$$ Then where does ...
0
votes
0answers
5 views

Autocorrelated, discrete, bounded and symmetric random walk with no edge attraction

I need to move over a discrete set of linearly organized.. let's say "Japan steps" $S=\{0,\dots,c\}, c \in \mathbb{N}^*$. My current position is given by $d \in S$. On each time step, I need to draw ...
0
votes
1answer
21 views

Probability distribution of $M_n = min(X_1 … X_n)$

I want to derive the distribution of $M_n=min(X_1 ... X_n)$ in another way than by a combinatorial analysis. Say we have $X_1...X_n$ represent $n$ draws without replacement from the numbers $1...N$ ...
0
votes
1answer
16 views

Derive $E(X^k)$ I need help with the substitution piece.

If $X\sim\mathrm{WEI}(\theta,\beta)$, derive $E(X^k)$ assuming $k > -\beta$. Note that $X\sim\mathrm{WEI}(\theta,\beta)=\dfrac{\beta}{\theta^\beta}x^{\beta -1}e^{-(x/\theta)^\beta}$ I know to ...
1
vote
1answer
61 views

Solve c value in $c \cdot (x+2y) \cdot e^{x+y} $

Today I started to look at previous exam questions, but I can't figure out the solution of one the questions. I hope someone could help me. In this question I have to find the c value: $$ ...
0
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0answers
11 views

Non-Linear System. Find the conditional expectation.

I've had my test for this course and I think I failed it again. The hardest part for me is findig the correct distributions. This is a test exercise I couldn't figure out or at least, I probably ...
1
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2answers
17 views

Minimum of the sample size estimator Bernoulli distribution

Given is a random sample $X_1 ... X_n$ from a $Ber(p)$ distribution. Consider the estimator $T = min\{X_1 ... X_n\}$. First, what is now the distribution of $T$? The minimum says that everything ...
0
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0answers
19 views

A conjecture on the connection between the difference of two independent Poisson random variables and their parameters.

Let $X$ and $Y$ be two independent poisson random variables with parameters $\mu$ and $\lambda$, respectively. Assuming that $\mu\geq\lambda$ , is it true that $P\left(X=Y-k\right)$ is decreasing in ...
0
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0answers
17 views

Expected value of a random variable and its square root

Can the expectation of a random variable be written in this fashion: $$E(\sqrt{X} + X) = E(\sqrt{X}) + E(X)$$ Thanks in advance
0
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1answer
30 views

Joint probability density for independent variables

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = e^{-x_i}$, for $x_i > 0$ for $i = 1,2$ (a) Find the joint probability density function ...
0
votes
1answer
39 views

Showing Convergence in Distribution for Conditional Random Variable

I am trying to prove the following: Let $X$ and $Y$ be random variables such that $Y | X = x$ ~ $N(0, x)$ with $X$ ~ $Po(\lambda$). Show that $\frac{Y}{\sqrt{\lambda}} \to N(0,1)$ in distribution as ...
-1
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0answers
104 views

Find measure such that…

I've a very concrete problem I can't solve. Consider the following function $k: [0,1]^2 \to \mathbb{R}:$ $$ k(x,y)=\begin{cases} 1 &\text{if } y > x \\ -1 &\text{if } x- \frac{1}{2} < ...
-2
votes
1answer
42 views

Let X and Y be two random variables with joint probability density function [on hold]

$f(x,y) = k(1+xy)$, $0<x<1$ and $0<y<1 $ (a) Find the value of $k$ such that $f(x,y)$ is a valid joint probability mass function. (b) Are $X$ and $Y$ independent? Justify your answer. ...
0
votes
1answer
41 views

How to find the expected cost of an exponential probability?

The length $X$ of of a call follows the exponential distribution with mean $2$ minutes. In dollars, the cost of of a call of $x$ minutes is $3x^2-6x+2$. Find the expected cost of a call? The addition ...
1
vote
2answers
34 views

Rayleigh Distribution: MLE biased?

This is most of an exam question I am doing for revision- some parts I have completed, others I am not sure about. We have $H$ the maximum height(depth?) of a river each year, modelled as a rayleigh ...
2
votes
1answer
84 views

What distribution has $X^n$ if $X$ is normal distributed?

Let $X$ be a random variable with mean $0$ and variance $\sigma ^2$, i.e. $X \sim \mathcal{N}(0, \sigma ^2)$.What is the distribution of $Y= X^n$, $n \in \mathbb {N}.$ ? I know what distributribution ...
0
votes
1answer
39 views

Given a CDF find the PDF

Let $$F(x) = 1 − \Bbb e ^{-x^3}; x > 0$$ be the cumulative distribution function of a continuous random variable $X$. (a) Find the probability density function of $X$. (b) Find the value of $c$ ...
0
votes
2answers
30 views

Let $X$ be a continuous variable with probability density function $kx(1-x)^2$ for $ 0<x<1$

Let $X$ be a continuous variable with probability density function $f(x)=kx(1−x)^2$ over $0< x <1$, zero otherwise. $(a)$ Find a value of $k$ so that $f(x)$ is a proper density. $(b)$ Find ...
-1
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0answers
21 views

Density function of $Y=X-\lfloor X \rfloor$

I am trying to solve the following assignment on the distribution of the rounding residuals of continuous random variables: Let $X$ be a continuous random variable with the density function $f_X(x)$ ...
3
votes
1answer
46 views

Proving a Variation of the the Central Limit Theorem

I am trying to prove the following: Let $X1, X2, . . .$ be positive, i.i.d. r.v.s with mean $\mu$ and finite variance $\sigma^2$, and let $S_n = \sum_{k=1}^{n} X_k$ , $n \ge 1$. Show that $\frac{S_n ...
2
votes
0answers
20 views

Distribution of $f(x,|h|)$, being $|h|$ rayleigh distributed

INTRODUCTION Let's supose we receive the following signal: \begin{equation} y[n] = hx[n]+W[n] \end{equation} where: $x[n] = Ae^{j2 \pi f_c t}$ is the transmitted signal $f_c$ is the carrier ...
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votes
0answers
35 views

How to compute P(Y>X)? [on hold]

Compute $P(Y>X)$ when $F_{X,Y}(x,y)= \begin{cases} 3\over4& \text{if }(0< x < 2) \wedge (0 < y < 2x-x^2)\\ 0 &\text{elsewhere}\end{cases}$ $F_X(x)=\begin{cases}\frac32 ...
1
vote
2answers
38 views

Confidence interval for Poisson distribution coefficient

This is an exam question, testing if water is bad - that is if a sample has more than 2000 E.coli in 100ml. We have taken $n$ samples denoted $X_i$, and model the samples as a Poisson distribution ...
-2
votes
0answers
57 views

How to solve for X^2-2Yx+Y=0? [on hold]

How can I solve for $x^2-2Yx+Y=0$? Note: Y is an exponentially distributed random variable with parameter lambda>0. The solution is the following: no real solution for $4Y^2-4Y<0$, so when ...
2
votes
1answer
29 views

The asymptotic equivalence of LR, Wald and score tests

Suppose that $Y_1, \ldots, Y_{n}$ are iid from a Bernoulli distribution with parameter $p$ and consider $H_0 : p = p_0\,.$ The test statistics are $$ T_W = \frac{n ({\widehat p} - p_0)^2}{{\widehat ...
0
votes
1answer
33 views

Simulating Random Vectors Based on Conditioning

I'm working on a project where I need to simulate random vectors $(Y, X_1,\dots,X_n)$ in order to understand the joint distribution $f(y,x_1,\dots,x_n)$. I wish to simulate enough random vectors so ...
2
votes
4answers
83 views

What does this definition mean: $F_Y(y) =P(Y<y)$?

I am doing calculations on $F_Y(y) := P(Y<y)$, but I am clueless as to what $P(Y<y)$ means. For instance the following question: Given function: $f_X(x)= 2\lambda x e^{-\lambda x^2}$ when $x ...
-3
votes
1answer
26 views

Mean and Variance of a Function of an Exponential distribution [on hold]

The question is: I know that the mean and variance of X are $1/4$ and $1/16$, but how would you find it for $Y$? I thought of using a moment generating function, but am confused as to how to do so. ...
0
votes
3answers
17 views

Combined Binomial Distribution Problem.

I have the following problem: 70% of women respond positively to a test, while only 40% of men do so. If 10 participants are selected (5 women and 5 men), what is the probability that only 1 man ...
2
votes
0answers
31 views

Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
1
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0answers
41 views

What is the product of two independent random variables (as mentioned below)?

Let $X$ and $Y$ be two random variables with: $\begin{equation} f_{X}(x) = \begin{cases} e^{-\lambda T} & \text{if } x = 0;\\ \lambda T e^{-\lambda T(1-x)} & \text{if } 0 < x \leq ...
1
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0answers
83 views
+300

Linear programming: constraints that depend on sign

Edit: following a comment, more detail and context, and removed lengthy confusing remains of previous edits I basically want to check whether there is a sequence $y_1,\dots, y_n \in (-\infty,0]$ that ...
0
votes
1answer
13 views

Probability of return with 7% error

I have a problem understanding the answer of the following problem: A recent audit by the IRS of the returns she prepared indicated that an error was made on 7% of the returns she prepared last ...
1
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0answers
17 views

How to solve for a Phase Function Cumulative Distribution function (CDF) calculation give a pdf …

I am attempting to solve for the CDF (more specifically the inverse CDF, but that is easy once I have the CDF) - Cumulative Distribution function given a Probability Distribution Function (pdf) and g ...
4
votes
1answer
28 views

Joint distribution of $(W(1),W(3),W(3)-W(2))$ for a Brownian motion $(W(t))_{t \geq 0}$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration. What is the joint probability distribution of ...
0
votes
0answers
23 views

Inverse Gaussian distribution - applying it to data

I'm trying to use three curves from a publication - all the information the publication gives is that the three curves follow an inverse Gaussian distribution with parameters $\mu = 73.9$ and ...
3
votes
0answers
21 views

Expected value of sorted subsequence

Consider the following discrete random variable: Given an array of size n containing random unique integers, what is the maximal length of a sorted subsequence from that array. What is the expected ...
5
votes
1answer
63 views

Checking the Lindeberg condition (central limit theorem)

Problem. Let $W_1, W_2,...$ be independent and identically distributed random variables such that $E(W_1)=0$ and $\sigma^2 := V(W_1) \in (0,\infty)$. Let $T_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n a_j ...
3
votes
1answer
31 views

Jacobian Transformation p.d.f

Suppose $X$ and $Y$ are continuous random variables with joint p.d.f. $$f(x,y) = e^{-y},\,\, 0<x<y <\infty$$ (a) Find the joint p.d.f. of $U=X+Y$ and $V=X$. Be sure to specify ...
0
votes
1answer
16 views

Linear combination of gaussian variables

If $X\sim N(0,\sigma_1^2)$,$Y\sim N(0,\sigma_2^2)$ and given that X,Y are independent random variable with normal distributions, then for the random variable $U=\alpha X+\beta Y\sim N(\mu,\sigma^2)$ ...
1
vote
1answer
44 views

Is it possible to determine if a process is random

Imagine the following experiment: someone is sitting behind the screen and calls out a sequence of numbers: "1! 3! 5! 3! 4! ...". Let's say he/she and I agreed beforehand that all numbers are ...
-1
votes
1answer
27 views

Probability Density Function of a velocity [closed]

Can you help me in computing a probability density function (pdf) of $$u(t) = 2 \sin \left( \frac{2 \pi t}{T}\right)$$ I ought to use the Binning approach:$$\int_{-\infty}^{+ \infty} f(u) \mathsf du ...
3
votes
3answers
56 views

Can a finite data set have all its values within $n$ standard deviations from the mean?

Aside from the trivial $(x,x,x,x,...)$ data set, is it possible to have all the elements of a data set within some $n$ standard deviations from the mean? What is the minimum possible value for $n$ ...
2
votes
1answer
43 views

Unable to understand what kind of pdf and its origin

I am facing difficulty in identifying how the formula given by Eq(2) in the paper Wen-Chi Tsai and Anirban DasGupta, On the Strong Consistency, Weak Limits and Practical Performance of the ML ...
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0answers
24 views

closed form p.d.f of euclidean norm of random variables

If $X\sim N(0,\sigma_1^2)$,$Y\sim N(0,\sigma_2^2)$,$Z\sim N(0,\sigma_3^2)$ and given that X,Y,Z are independent random variable with normal distributions, then the random variable $U=\sqrt{X^2+Y^2}$ ...
0
votes
1answer
25 views

Uniform distribution on sphere

Let $U = (u_1, u_2, u_3)$ is random vector uniformly distributed on unit sphere $S^{2} \subset \mathbb{R}^3$. Are $u_1, u_2, u_3$ mutually independent ? I guess not, but I have no idea to prove it.
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0answers
31 views

Expected probability maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like $$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$ Here $f(k,x)$ is actually a probability coming from a ...
0
votes
0answers
32 views

Expected Value of Product of Normal and Log Normal Distribution

Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} & & ...
0
votes
0answers
31 views

What is the probability of a collision (birthday problem) using a Zipf distribution?

Assuming a random number generator gives one of n numbers using a Zipf distribution rather than a uniform distribution, what is the probability of each number being a number that was already generated ...