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

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0
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0answers
8 views

How to interpret a p-value that's significant from Fisher's Exact test

Given a binomial distribution with p=.03, n=902, the $.025$ and $.975$ quantiles are $17$ and $38$ respectively. I interpret ...
0
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0answers
20 views

An airline company annually gives $5000$ vouchers to its clients. In the previous year, $67 $% of the clients that received a voucher redeemed it.

(a) Let X be number of clients that will redeem the voucher this year. Suggest a distribution for X. State any assumptions that you need to make. (b) If the average cost of each voucher is $ $125$, ...
1
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1answer
20 views

Determining Probability Generating Function from Probability Mass Function and Convergence

I am trying to solve the following: Suppose $X_{nk}, k=1,2,\ldots,n, n≥ 2$ are i.i.d. random variables $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
2
votes
1answer
26 views

Application Problem: Conditioning Poisson Process

I am trying to solve the following application problem: There are $n$ components with independent lifetimes which are such that component $i$ functions for an exponential time with rate $\lambda_i$. ...
0
votes
2answers
31 views

probability density functions and cumulative distribution function

Suppose $X$ is an absolutely continuous real random variable, (that is, there exist a non-negative integrable function $f$, such that $\int_\mathbb{R} f=1$ and for every interval $I\subseteq ...
-2
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1answer
16 views

Statistics probability [on hold]

A coffee shop sells 4 sizes with 4 different varieties. Customers can choose to add one or more syrups that come in 4 flavors. How many different coffees drinks can be made?
0
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1answer
14 views

When do I use Law of total variance?

For example, at the beginning of doing this problem (http://math.illinoisstate.edu/krzysio/3-6-10-KO-Exercise.pdf), I was thinking of using $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, ...
2
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0answers
11 views

Is the product of two sub-Gaussian random variables a sub-Gaussian random variable?

If not, is there any way to make it hold? Note: the random variable $x$ is called $σ^2$-sub-Gaussian if $E[e^{tx}]≤e^{t^2σ^2/2}$.
0
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0answers
29 views

Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.

Assume a die is rolled repeatedly. Find the markov/transition matrix $P$ for the random variable $X_r$ = the time until the next six at time $r$. My solution was: For $i,j \geq 0$, $P$ is given ...
1
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0answers
32 views

Application of Slutsky's Theorem to the Convergence of Sum of R.V.

Let $X_1, X_2,…, X_n$ be i.i.d. $U(−\theta,\theta)$. Show that $Z_n \to N(0,\sqrt{\frac{5}{9}}$ in distribution, where $Z_n ...
0
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0answers
24 views

Find/estimate variance

Let $w_{11},\ldots , w_{nm}\in [0, 1]$ be a set of constants and $H_1(t), \ldots , H_m(t)$ be some cumulative distribution functions (CDFs). Consider a sample of independent random variables $\xi _1, ...
1
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1answer
14 views

Find a distribution for this plot

Please help me find a formula that fits the distribution. It does not need to be exact, a simple approximation would suffice. Bonus points if you can tell me which predefined distribution in the ...
0
votes
1answer
20 views

Cumulative Distribution Funciton to pmf

I am still quite new to cdf and pmf. When we only have pmf for x = 1, 2 and 4 , how should I understand the corresponding cdf as in the pmf for x = 3 doesn't exist. Also I tried to draw the piecewise ...
1
vote
1answer
26 views

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = 1$, for $0 < xi < 1$ for $i = 1, 2$.

Find: (a) $E(X_1 X_2)$, and (b) $Var(X_1 X_2)$. Isn't (a) = zero, since this are independent? How do I go about (b)
1
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1answer
34 views

Let X and Y be a random variables with $E(X) = 5$, $Var(X) = 30$, $E(Y ) = -􀀀5$, $Var(Y ) = 10$ and $Cov(X, Y ) = 7$

(a) Find $E(2X-3Y+1)$. (b) Find $E((X-2Y)^2)$. (c) Find $Var(3X-Y+pi)$ First I found $E(X^2)$ and $E(Y^2)$ using the given values for (a) I have $2E(X)-3E(Y)+1$ for (b) I come up with: ...
1
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1answer
24 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
0
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2answers
18 views

Operations with probability distributions

I had an idea that passes by declaring a new type of computer variable (like Integer, Double, etc.) that represents a statistical probability distribution (PDF), for that I would need to define the ...
1
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0answers
17 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
1
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2answers
18 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!
1
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0answers
15 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 ...
1
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0answers
24 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
25 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
18 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 ...
2
votes
1answer
65 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
16 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
vote
2answers
20 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 ...
1
vote
1answer
28 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
21 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
votes
1answer
35 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
41 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
108 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
50 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
42 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
37 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
2answers
98 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
42 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
35 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 ...
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
21 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
38 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
42 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 ...
-1
votes
0answers
64 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
84 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
27 views

Mean and Variance of a Function of an Exponential distribution [closed]

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
18 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
32 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
vote
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
vote
0answers
114 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 ...