Linked Questions

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0answers
56 views

Proof that the sample mean is normally distributed [duplicate]

I am studying the book of Larsen and Marx and stumbled upon I can prove that $\bar{Y}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}$ and $Var(\bar{Y})=\frac{\sigma ^{2}}{n}$ but how would I go to show that $\bar{...
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0answers
50 views

Sum of 3 correlated normal random variables [duplicate]

I know that sum of 2 correlated normal random variables are written as $X+Y\sim N(\mu_x+\mu_y,\sigma^2_x+\sigma^2_y+2\rho_{xy}\sigma_x\sigma_y)$, I am now wondering how is 3 correlated normal random ...
1
vote
0answers
41 views

Gaussian random variables [duplicate]

Can some one help me point in the right direction or point to some resources that will help me prove that sum of two jointly distributed Gaussian r.v. with a given correlation coefficient is also a ...
16
votes
2answers
25k views

Linear combination of two dependent multivariate normal random variables

Suppose we have two vectors of random variables, both are normal, i.e., $X \sim N(\mu_X, \Sigma_X)$ and $Y \sim N(\mu_Y, \Sigma_Y)$. We are interested in the distribution of their linear combination $...
8
votes
1answer
12k views

The product of two lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas. Consider the corresponding log-normal random variables: $...
0
votes
3answers
8k views

Standard deviation of the sum of two normally distributed random variables

$X\sim N(52,6)$, $Y\sim (40,8)$. What's the standard deviation of $Z=X+Y$? I'm considering to transform the linear relationship to matrix form $$Z=\begin{pmatrix} 1& 1\\ \end{pmatrix}\begin{...
3
votes
2answers
7k views

Sum of Gaussian is Gaussian?

As a newbie in probability, I am recently cleaning my understandings about Gaussian distribution. I know that If $X$ and $Y$ are jointly Gaussian, then $aX+bY$ ($a$ and $b$ are both constant) is ...
5
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1answer
9k views

How to compute a probability threshold for a linear combination of two variables ~ N(0,1)?

I have a variable which is a linear combination of two other variables, each one following an N(0,1) distribution. I need to compute the threshold of the distribution of this combination variable (to ...
5
votes
1answer
2k views

Joint distribution of sum of independent normals

Suppose we have three independent normally distributed random variables $$ X_0 \sim \mathcal{N}(\mu_0, \sigma_0^2), $$ $$ X_1 \sim \mathcal{N}(\mu_1, \sigma_1^2), $$ $$ X_2 \sim \mathcal{N}(\mu_2, \...
7
votes
1answer
349 views

Generate identically distributed dependent normal random numbers with prespecified sum

How do I generate $n$ identically distributed but not independent normal random numbers such that their sum falls within a prespecified interval $[a,b]$ with probability $p$? (This question is ...
1
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2answers
1k views

Is the joint distribution of two linear combinations of Gaussians still a multivariate normal?

Suppose I have $\textbf{X}$ ~ $N(\textbf{0},\Sigma)$, and I'm considering two different linear combinations, $a^* X$ and $b^* X$, which we suppose are uncorrelated. I understand that linear ...
1
vote
1answer
146 views

Flying Bomber aircraft through SAM sites - combining normal distributions

I have been puzzling over this for days but I don't think this was covered at school We simultaneously fly a known number of bomber aircraft, K, through three sequential batteries of surface to air ...
1
vote
1answer
25 views

Marginal distributions of two linear transformations of two correlated Gaussian (Normal) distributions

Considering this entry the distribution of the sum of non i.i.d. gaussian variates is also gaussian. $$ \begin{align*} V = aX + bY &\sim N(a\mu_X + b\mu_Y,\; a^2\sigma_X^2 + b^2\sigma_Y^2 + 2ab\...
0
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0answers
29 views

How to show via Delta method that the Linear Taylor series expansion of a normal random vector results in NORMAL DISTRIBUTION [duplicate]

How can it be proved using the delta method that the Linear Taylor series expansion of a normal random vector containing independent but NOT identically distributed elements results in a random ...