-1
votes
0answers
33 views

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

I have $X,Y$ iid Normals $N(0,\sigma^2)$ What is the distribution of $X+Y+YX$? Thnks a lot!
2
votes
0answers
15 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
46 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
34 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
55 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
49 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
42 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
181 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
64 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
78 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
17 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
72 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
50 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
72 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
60 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
21 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
34 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
221 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
47 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
29 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
43 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
42 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
261 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
140 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
61 views
0
votes
1answer
101 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
60 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
36 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
69 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
88 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
529 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
68 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
83 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 ...
2
votes
1answer
102 views

Weighted integral of random variables

Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random ...
0
votes
0answers
61 views

iterative transform of standard normal random variable

Given a discrete series of random variable $n(i)$ that each element follows the standard normal distribution $N(0,1)$, another series is defined iteratively as: $$u(i+1)=au(i)+bn(i)$$ where ...
0
votes
1answer
96 views

Normal random Vector

Question: Prove that linear functions of the form $\bar{y}=\bar{b}+\mathrm{B}\bar{x}$ are normal random vectors provided that $\bar{x}$ is a normal random vector. Find $E(\bar{y})$ and $V(\bar{y})$. ...
0
votes
0answers
55 views

Is there a worded problem that results in this transformation of a Normal RV?

I am familiar with worded questions that require the sum of normal RVs to be found. The width, in mm, of a book is distributed $X\sim N(10,\,2)$. Find the probability that the total width of 5 ...
0
votes
0answers
41 views

Formal definition of the transformation of a lognormal variable.

I'm just after a clarification on the transformation of a lognormal distribution $X$. Let's say we have $X = x*e^{N(\mu,\sigma^2)}$ How exactly would we transform this into a standard lognormal? Is ...
1
vote
1answer
223 views

Computing the expected value of a matrix?

This question is about finding a covariance matrix and I wasn't sure about the final step. Given a standard $d$-dimensional normal RVec $X=(X_1,\ldots,X_d)$ has i.i.d components $X_j\sim N(0,1)$. ...
1
vote
0answers
74 views

P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined $b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
4
votes
2answers
92 views

Central Limit Theorem. How to apply to the task.

The research showed that the probabilities of 3, 4, 5, 6 and 7 cars broken on one day are 0.3, 0.4, 0.2, 0.08, 0.02 respectively. If 221 car broke in 50 days, does it show that more cars break than ...
1
vote
1answer
82 views

Simple question on random variables and statistics

Let X1 and X2 be 2 random variables. X1 = 20. X2 = 30. Each of those has a standard deviation of 5. If the random variables were normally distributed, what is the probability of getting such a ...
0
votes
1answer
52 views

Homework Help. Probability Density Functions.

$X$ is $N(10,1)$. Find $f(x|(x-10)^2 < 4)$ This is a homework question. I can only figure out that X is normally distributed with mean 10 and variance 1. Can you please explain what is meant to ...
6
votes
0answers
200 views

What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
2
votes
3answers
5k views

Standardizing A Random Variable That is Normally Distributed

To standardize a random variable that is normally distributed, it makes absolute sense to subtract the expected value $\mu$ , from each value that the random variable can assume--it shifts all of the ...
0
votes
1answer
109 views

How to prove that $Y=\ln(X)$ approximately Normal when $X$ is a Normal random variable with $\mu\gg\sigma$

I wanted to prove that PDF of $Y=\ln(X)$ tends to a Normal distribution with $\mathcal{N}(\ln(\mu_{x}),\sigma^{2}_{y})$ when $X\sim\mathcal{N}(\mu_{x},\sigma^{2}_{x})$. It is also important to note ...
0
votes
0answers
56 views

Using an appropriate hypothesis to test whether two means are different

Manager examined potential differences between two models of bicycles. The mean life of the bicycles is of primary concern. The followings table provides the available date which measured in ...
-1
votes
3answers
224 views

Variance of transformed random variable

The relationship of two random variables is given by $$ X = \Phi(Y) \Leftrightarrow Y = \Phi^{-1}(X),$$ where $\Phi(\bullet)$ is the standard normal cdf and $\Phi^{-1}(\bullet)$ the inverse of the ...