0
votes
0answers
12 views

Generate Correlated Normal and Log-Normal Random Variable

The standard approach for generating two normally distributed random variables some with correlation $\rho$ is explained here: Generate Correlated Normal Random Variables. Now let $X,Y$ be normally ...
0
votes
1answer
38 views

Convergence sequence of random variables

I have this problem about a sequence of normals. $(X_n)_{n\geq 0}$ is defined as $$X_{n+1}=aX_n+U_{n+1}$$ $X_0=0$, where $(U_n)_{n\geq1}$ is a sequence of i.i.d random variable normally distributed ...
3
votes
3answers
48 views

Transformation(?) of Random Variables

There are two independent Gaussian R.Vs: $U:N(-1,1)$ and $V:N(1,1)$ How do I go about finding the PDF of the following transformations? X = U+V T = (U+2V, U-2V) W = U (with 50% chance), V (with ...
2
votes
1answer
13 views

Distribution of a function of a normally distributed variable

Let's say you have a random variable $X$, which is normally distributed according to $X \sim \mathcal{N}(1,2)$. With $1$ being the mean and $2$ being the variance. Now let's say that there is another ...
0
votes
1answer
21 views

Normal Random Variables

Let Z1 and Z2 be independent standard normal random variables. What is the probability that the minimum of Z1 and Z2 will be greater than 1.0? How do I go about this when I have no values? Is the ...
2
votes
1answer
34 views

$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), Z_2:=\sqrt{-2\log X} \sin(2\pi Y)$ independent and normal

I am looking for a nice proof of the following statement: If $X,Y\sim U(0,1)$ are two independent uniformly distributed random variables, then $$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), \quad ...
0
votes
0answers
13 views

Two i.i.d Rvs (Gaussian)

Q: You have two i.i.d Rv's X~N(0,1) Y~(0,1). Let Z=(X+Y)^2. a) Find the mean on Z i.e E[Z}. b) Find Corr(X,Z) & Corr(Y,Z). c) Determine if Z & Y are uncorrelated. Ans: Finding E[Z] was ...
1
vote
1answer
45 views

Convergence in probability of iid normal random variables

Let $X_1, X_2,\ldots$ be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1) $X_n \to X_1$ in law, (2) $X_n \not\to X_1$ in ...
0
votes
0answers
14 views

Controlling a random variable

I've got a system the output of which is a random variable with a certain distribution, which for the purpose of this discussion can be assumed to be normal. The input variable is voltage. The ...
3
votes
0answers
34 views

Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that ...
0
votes
1answer
43 views

Finding probability density function of a linear combination of mutually independent normal random variables

I'm finding the probability density function of the random variable U defined in the following manner: $$U=\frac{1}{2}(Y_1+3Y_2)$$ CORRECTION: The line above is supposed to be ...
-2
votes
1answer
53 views

What's the pdf of $Z=X^2 +2X$ if $X$ is a standard normal? [closed]

Le be $X$ distributed as a standard normal. What is the density function of $Z=X^2 +2X$? Thanks for your help
0
votes
3answers
39 views

Independent variables, normal distribution, pdf

I have independent variables $ X_1, X_2,\ldots,X_n $ with normal distribution on range $ [0,1] $ . Next, variables $ Z_i $ are created according to this formula $ Z_i = - \frac{1}{\lambda} \ln(1-X_i) ...
1
vote
0answers
123 views

Limit of sequence of integral related i.i.d. observations

Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat ...
-1
votes
1answer
38 views

If speeds of two cars are Normal RV s, what is the distribution of the distance between them?

The speeds of two cars are random variables that follow $N(\mu_1,\sigma_1)$ and $N(\mu_2,\sigma_2)$ distributions.They start simultaneously. Let X be the distance between them after m hours. (Note ...
3
votes
0answers
171 views

PDF of X +Y + X* Y, when X and Y are independent Normal [closed]

I have $X,Y$ iid Normals $N(0,\sigma^2)$ What is the distribution of $X+Y+YX$? Thnks a lot!
2
votes
0answers
25 views

Variance of a Population of Two Indpendent Random Variables

I have a question regarding a problem I'm looking at out of personal curiosity. Here is the basic setup of the problem: There is a population that contains half of type A, and half of type B. The ...
0
votes
1answer
22 views

Condition on variable to make events independent

where, $$n_1,n_2,...,n_M \sim N\left(0,\frac{N_0}{2}\right) $$ how the condition on n_1 makes the events independent ? what is "n_1=n"
1
vote
0answers
52 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
0
votes
1answer
50 views

variance of a random variable

If $X_1, X_2 , ....., X_n$ iid $N(0,1)$ , and $S^2$ was defined as the population standard deviation we are to find the variance of $S^2$ I want to know the distribution in order to find the ...
0
votes
2answers
140 views

moment generating function technique

If $X$ was a random variable with a distribution $\mathrm{Normal} ( 0, 1 )$, using moment generating function technique we have to show that $Y= X^2$ has the Chi-square distribution with $1$ degree of ...
1
vote
2answers
52 views

Sampling from a Normal Distribution

If I am sampling randomly from only the -sigma to +sigma interval of a normal distribution and rejecting all other numbers, does it imply that the probability density changes? If so, by what degree? ...
0
votes
0answers
56 views

Absolute value of the Fourier Transform of Gaussian random variable

Assume you have a normally distributed random variable $x$ with zero mean $\mu$ and standard deviation $\sigma$. Now you take the Fourier transform of it. The resulting complex random variable ...
0
votes
2answers
540 views

Are any linear combination of normal random variables, normally distributed?

It is easy to show that if we have n independent normally distributed random variables, then a linear combination fo them ar normally distributed. It is also said that if (x1,x2,..,xn) is ...
0
votes
0answers
90 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
1
vote
1answer
85 views

help with Borel Cantelli lemma

There is a sequence of random variables $X_1,X_2,...$ For each i $X_i$ ~ $Normal(0,1)$ Is $ \frac{X_n}{n} \rightarrow 0 $ almost surely? Is $ \frac{X_n}{lnn} \rightarrow 0 $ almost ...
0
votes
0answers
18 views

How to generate normally distributed random numbers? [duplicate]

Is there any function that can generate normally distributed random numbers?
0
votes
0answers
19 views

Sampling distribution with large sample size

As the sample size $n$ of a sampling distribution of sample means increases, the distribution becomes more normal. But if $n$ were the same size as the (finite) population, the "sampling" distribution ...
0
votes
2answers
98 views

Adding two normal distribution

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. And Suppose that $Z \sim N(1, 2^2)$ and is independent of all $X_i$. Define $Z_i = Z + X_i$ for $i = 1, ...
0
votes
2answers
56 views

Central Limit Theorem Application on Multivariate Normal

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. What is the distribution of $\overline{X} = \frac{1}{3}(X_1+X_2+X_3)$? I don't quite understand how to ...
1
vote
1answer
109 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
1
vote
1answer
95 views

Showing that two Gaussian processes are independent

Say that $Z_t = (X_t, Y_t)$ is a 2-dimensional Gaussian process (by definition, it means that the random vector $(X_{t_1},Y_{t_1},...,X_{t_n},Y_{t_n})$ is a Gaussian random vector for all $t_1 ...
0
votes
1answer
25 views

Probability question of independent random varaibles

Let $X\sim \mathcal{N}(6,1)$ and $Y\sim\mathcal{N}(7,1)$ be two independent normal variables. Find $Pr(X>Y)$. the answer is $0.2389$ but I do not know how to do it.
0
votes
1answer
38 views

Error function property

I have a question regarding a property of the error function. Is $k\cdot\text{erfc}(-x) = 1-k\cdot\text{erfc}(x)$ for all real $x$ for any $k$?
1
vote
1answer
281 views

Sum of two independent normal distributed random variables

If $X_i$, $i =1,2$ are independent and have normal distribution with mean $0$ and variance $\sigma_i ^2$. Show that $X_1 + X_2$ has a normal distribution with mean $0$ and variance $\sigma_1^2 + ...
2
votes
1answer
58 views

change of variable in normal distribution

The normal distribution of random variable $x$ is $$p(x)=Norm_x[Ay+b,\Sigma]$$, the mean $\mu=Ay+b$ is a function of another variable $y$. My problem is how to derive the normal distribution of $y$, ...
1
vote
1answer
35 views

Confidence interval and normal distribution

For question (a), is the answer 0.7143? For question (b), is the answer 10.85 and 11.95 ?
1
vote
1answer
52 views

change of unit normally distributed random variable

Assume that $X_{1}$,$X_{2}$,$X_{3}$ are independent continues random variables with $\mathcal{N}(30,12)$, what is the normal distribution of $X_{average}$ (average of $X_{1}$,$X_{2}$,$X_{3}$) ...
0
votes
1answer
45 views

Normal approximations and confidence interval

Let $X$ be the number of times that a fair coin, flipped $40$ times, lands heads. $\text{(a)}$ Find the probability that $X=20$. $\text{(b)}$ Use the normal approximations and then ...
1
vote
1answer
309 views

Probability: Normal Distribution

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability $0.95$. Approximate the probability (by a normal distribution) that at most $10$ of the ...
0
votes
1answer
158 views

Probability: normal distribution and standard normal random variable

Let $X$ follows the normal distribution $N(1,9)$. Find $\text{(a)}$ $P(X\le1.4).$ $\text{(b)}$ $P(X\le-1.22).$ $\text{(c)}$ Hence find $P(-1.22\le X\le1.4).$ For $\text{(a)}$, is ...
1
vote
1answer
69 views

Generating 2D random vector from 4D covariance matrix

I have such covariance matrix $C$: ...
0
votes
1answer
112 views

Joint Probability Distribution of a Gaussian Random Variable Correlated with a Gamma Random Variable

I want to know if the joint PDF of a Gaussian RV correlated with a Gamma RV can be found in closed form. The correlation is known.
0
votes
1answer
73 views

$\sum(y_i-\bar{y})^2$ can be written in the form $\sigma^2 X'AX$ where $X\sim N(0,1)$. What is $A$?

Random sample $Y_1,\dots, Y_n$ of size n from a univariate normal population with ($\mu, \sigma^2$). Let $\bar{y}=\frac{1}{n}\sum Y_i$. $\sum(y_i-\bar{y})^2$ can be written in the for $\sigma^2 X'AX$ ...
0
votes
0answers
41 views

Expected value of the sum of r.v. with parameter-dependent mean

I have two r.v. $X(p)$ and $Y(p)$ whose mean $\mu$ depends on a parameter $p$, while the $\sigma$ is given for both variables. For the sake of this problem, let's say the variables are normally ...
2
votes
1answer
79 views

Given a covarince matrix, generate a Gaussian random variable

Given a $M \times  M$ desired covariance, $R$, and a desired number of sample vectors, $N$ calculate a $N \times M$ Gaussian random vector, $X$. Not really sure what to do here. You can calculate ...
1
vote
0answers
93 views

Calculate the variance from a function of normal random variable

I am new to the topic that I found difficulty for the question: Given the function $g(x) = e^{-X}$, $X \sim N(0,1)$, calculate the variance of $g(x)$. I know the answer is $e(e-1)$. But I don't ...
3
votes
2answers
651 views

Sums of Products of Two Normal Variables

Suppose that $X_1 ,\ldots,X_n,Y_1,\ldots,Y_n$ are all independent normal random variables with different means and variances. What is the PDF of the following random variable? ...
1
vote
1answer
101 views

Correlation of sums of correlated variables

I'm trying to work out an expression for a correlation of the weighted sums of two r.v.'s with a third r.v. To be precise, I have a trivariate normal distribution: $$\{X,Y,Z\}\approx ...
1
vote
1answer
91 views

distribution of maximum of $n$ Pearson correlations

$\mathbf{x}=[x_1,x_2,...,x_m]^{\top}$ is a vector of length $m$ and $\mathbf{y_1}, \mathbf{y_2}, ..., \mathbf{y_n}$ are similarly $n$ vectors of length $m$. If the elements of $\mathbf{x}$ and ...