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

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17 views

How do simulate a biased normal distribution using dice

I saw this which suggests 5 dice is enough to simulate a normal distribution. http://www.johndcook.com/blog/2009/02/10/rolling-dice-for-normal-samples/ How can you simulate a biased normal ...
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2answers
24 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
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0answers
11 views

Family of parameterized distribution functions with separable inverse

I am looking for parameterized families of distribution functions that have a separable inverse. For example, the inverse of the exponential distribution is $H^{-1}(p;\gamma)=\frac{1}{\gamma}ln(1-p)$ ...
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1answer
32 views

Statistics - normal distribution problem

Two random variables $X$ and $Y$ are i.i.d. normal$(\mu, \sigma^2)$. If $P(X > 3) = 0.8413$ find $P((X+Y)/2 > 3)$. The result must be exact number, so normal distribution parameters are ...
2
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1answer
25 views

Inequality for the derivative of a density of a random variable convolved with a normal r.v.

I have a question about the following proof. The statement is: Let $X$ be a random variable and $Z_\tau \sim N(0,\tau)$ be an independent random variable. Then $Y_\tau := X + Z_\tau$ has a ...
2
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1answer
30 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
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1answer
32 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
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0answers
16 views

(Almost) Gaussian distribution

I need to find a distribution for random variable $\boldsymbol \epsilon$ goverened by parameter $\alpha >0$, such that: for any given $\boldsymbol \pi \in \mathbb [0, 1]^M: (\boldsymbol ...
3
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0answers
22 views

Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024x1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised ...
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2answers
29 views

Is my probability reasoning here correct?

Sheldon Ross theoretical exercice A jar contains $n$ chips. Suppose that a boy successively draws chips from the jar, each time replacing the one drawn before drawing another. The process continues ...
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0answers
22 views

Find the Lipschitz constant of a multi-variate Gaussian density function

I would like to find the Lipschitz constant of a multi-variate Gaussian density function: $$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^{k}|\boldsymbol\Sigma|}} ...
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votes
1answer
12 views

Density of a distribution function at upper bound [on hold]

Consider a strictly increasing continuously differentiable distribution F with support on $[a,b]$. Let $f$ be the pdf of $F$. What can we say about $f(b)$? Under what conditions is $f(b)>0$? ...
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0answers
20 views

A Question about the Kurtosis

Problem: Show that if a binomial distribution with $n = 100$ is symmetric, its coefficient of kurtosis is 2.9. Answer: First, I am interpreting the term symmetric to mean that $p = q = \frac{1}{2}$. ...
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2answers
24 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
1
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1answer
15 views

example of convergence in distribution but not in probability

While I was looking for an example of a sequence of random variables which converges in distribution, but doesn't converge in probability, I have read that it should be enough to consider a sequence ...
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0answers
6 views

generate binomial binary data

How does one generate correlated binomial data when one is given marginal probabilities of each and also the correlation coefficient. The following code in SAS for example works best when we want ...
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0answers
32 views

Find the expected value .

I am having a trouble in in the following questions. I would like if some one could help me. Thank you so much for your time. Simplify $$K=\int_0^T\left[\int_0^t e^{-\alpha(t-s)}dJ(s)\right] dt $$ ...
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1answer
25 views

Prove that the increments of the Brownian motion are normally distributed

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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0answers
14 views

Optimal decision for sampling a distribution.

I was wondering which probability distribution is best sampled with $\pm\alpha^n, n\in\{1,2,\cdots\}$ for various values of alpha. Sampling means to pick the one which is closest, store the sign and ...
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1answer
27 views

Failed to understand basic approach of mathematical modeling in research ideas [on hold]

I am very new to mathematical modeling. I study different probabilistic approaches but fail to understand how it can be implemented over any research activity. Is there any set of rules to apply for ...
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0answers
22 views

Closeness in distribution implies closeness in statistics?

I am aware that convergence in distribution does not necessarily imply convergence in the mean. I browsed through some examples of statistical distances here ...
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0answers
13 views

Copula vs Exprimental Copula

I have read some texts about finding/approximating copulas for a given sample based on known (famous) copulas. My question is: when we have the experimental CDF of (X, Y), why we should try to find a ...
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1answer
26 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 ...
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0answers
49 views

Select the correct probability distribution for the given problem [on hold]

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 ...
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1answer
34 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
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1answer
39 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 $\lambda_i$. ...
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2answers
32 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 ...
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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?
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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
12 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 $E[e^{tx}]≤e^{t^2σ^2/2}$.
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1answer
38 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 ...
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0answers
37 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
26 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 ...
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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
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1answer
27 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)
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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
25 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 ...
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2answers
19 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 ...
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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 - ...
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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!
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0answers
18 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 ...
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0answers
29 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 ...
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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 $1...N$ ...
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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: $$ ...
2
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1answer
29 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 ...
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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 ...
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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 ...
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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