# Expected value and variance from random variable

Given the following

$$Y_1 \sim \mathcal{N}(μ, σ^2 )$$

and

$$Y_2=α+βY_1+U \;where \; Y_1 \;and \;U\;is\;independent\;and\;U∼\mathcal{N}(0,v^2)$$

Let $$μ=350$$ and $$σ^2 =12365$$

How would i calculate the expected value and variance from $$Y_2$$ ? And how can i find the distribution of it?

Pretty much by definition of expected value: $$E(Y_2) = E(\alpha + \beta Y_1 + U)\\ = E(\alpha) + E(\beta Y_1)+ E(U)\\ = \alpha + \beta E(Y_1) + E(U)$$

And variance: $$var(Y_2) = E(Y_2^2) - E(Y_2)^2$$ We need to find $$E(Y_2^2)$$, which is

$$E\left[(\alpha + \beta Y_1 + U)^2\right]\\ = E\left[\alpha^2 + \alpha \beta Y_1 + \alpha U\\ + \alpha \beta Y_1 + \beta^2 Y_1^2 + \beta Y_1 U\\ + \alpha U + \beta Y_1 U + U^2\right].$$

The rest you can probably do with linearity and some arithmetic, I hope this helps.

Edit: Most of the work comes from linearity of the expect value function.

• I updated my post with more information if it makes any difference, cause im not sure i understand Commented Mar 20, 2021 at 16:57
• It seems you added the specific values, but that does not change how we approach the problem. We can solve it abstractly, and then plug in the numbers in the end.
– JLMF
Commented Mar 20, 2021 at 17:33
• I see but im not sure how to calculate the expected value and variance using the formulas you gave. Commented Mar 20, 2021 at 17:55
• What is the distribution in this case @BruceET Commented Mar 21, 2021 at 4:47

Because of independence:

$$m_Y=E(Y_2)=\alpha +\beta \mu$$ $$\sigma^2_Y=var(Y_2)=var(\alpha)+var(\beta Y_1)+var (U)$$

$$=\beta \sigma^2+ v^2$$

$$Y_2$$ ~ $$N(m_Y,\sigma^2_Y)$$.

You need values for $$\alpha$$, $$\beta$$ and $$v^2$$.

• I think alpha and beta can just be variables and not concrete values Commented Mar 20, 2021 at 21:17
• In that case you can't get a numerical result for mean or variance. Commented Mar 20, 2021 at 21:39
• So what would the expected value and variance be in this case without concrete values? Sorry im not that good at this Commented Mar 20, 2021 at 21:43
• The formulas given in the answer are all you can get. Commented Mar 20, 2021 at 21:47
• If i have to split it up into $E(Y_2) = something$ and $Var(Y_2) = something$ how would i do it from the formulas you've given? and i guess $Y_2$ ~ $𝑁(𝑚_𝑌,𝜎^2_Y)$ is the distribution but what is $m_Y$ ? and can it be written in another way? Commented Mar 20, 2021 at 22:31

You still haven't given values for $$v, a, b,$$ so I have supplied some.

The distribution of $$Y_2$$ is normal.

In my simulation in R below, I have used a million iterations, so the sample mean and SD of $$Y_2$$ should agree with the respective population values to several significant digits--I guess, good enough to find mistakes in applying the formulas.

set.seed(2021)
y1 = rnorm(10^6, 350, 111.198)
u =  rnorm(10^6, 0, 50)
a = 10;  b = 50
y2 = a + b*y1 + u
summary(y2);  sd(y2);  var(y2)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
-8248   13765   17511   17513   21263   43899
[1] 5566.788  # aprx SD(Y2)
[1] 30989134  # aprx Var(Y2)


Here are histograms of the simulated distributions along with the corresponding normal density curves.

R code for figure.

par(mfrow=c(1,3))
hist(y1, prob=T, col="wheat")
hist(u, prob=T, col="wheat")
hist(y2, prob=T, col="skyblue2")

• You say in a comment that you don't follow how to calculate some of the results for $$Y_2.$$ That is not a comment that prompts specific explanations.
• Do you mean the distribution of $Y_2?$ That's also normal. The purpose is to find its mean and variance. Commented Mar 21, 2021 at 5:48
• I mean in terms of notation, how would it be written, if im following @JlF 's advice? $X \sim something$ also is it possible to find the PDF? Probability density function? Commented Mar 21, 2021 at 17:47
• $Y_2 \sim \mathsf{Norm}(\mu, \sigma),$ where my simulation shows $\mu\approx 17513, \sigma\approx 5566.788,$ and you can get exact values of $\mu$ and $\sigma$ using the other answers. Once $\mu$ and $\sigma$ are known, plug them into the general formula for a normal density function. (This density function is not a secret; most prob and stat books have it.) Commented Mar 21, 2021 at 19:04