Sums Of RVs

Question 1

What is E (sigma i=1 to n Xi)

Answer 1

E(sigma i=1 to n Xi) = sigma i=1 to n E(Xi) e.g E(X1+X2)= E(X1) + E(X2)

Question 2

What is Var(sigma i=1 to n Xi)

Answer 2

Var(sigma i=1 to n) = sigma i=1 to m Var(Xi) + 2nested sigma 1<= I < j <= n Cov(Xi,Xj)
E.g var(X1+X2) = Var(X1) + Var(X2) + 2Cov(X1,X2)

Question 3

What is convolution formula

Answer 3

Convolution formula is:
If X and Y are independent continuous RVs, then W = X+Y has PDF
fw(w) = integral infinity down to -infinity fX,Y(x, w-x) dx = fx(x) * fY(w-x)

Question 4

What is convolution formula for 2 positive RVs

Answer 4

Convolution formula for 2 positive RVs is:
fx(x) = integral w down to 0 fX,Y(x,w-x) dx
And if X,Y are independent fX(x) * fY(w-x)

Question 5

If X1,…,Xi are I.I.d with X~N(mu, sigma^2), then what is sum Xi 1 to n and 1/n sum Xi i =1 to n

Answer 5

If X1,…,Xi are I.I.d with X~N(mu, sigma^2), then:
Sum Xi i = 1 to n Xi ~ N(nmu, nsigma ^2)
1/n sum Xi i =1 to n = Xbar ~ N(mu, sigma^2 /n)

Question 6

What does central limit theorem state

Answer 6

Central limit theorem states that:
If X1,… are i.i.d variables with finite expectation and variance then:
P[ (Sn - n*mu)/root(n * variance) < a] tends to PHI (a) as n tends to infinity where Sn is sum of X1,…,Xn and PHI(a) is CDF at a of a standard normal distribution

Question 7

What does CLT tell us and what can we do

Answer 7

CLT tells us that for large n :
(Sn - nmu)/root(nvariance) ~ N(0,1) approx
And we can do:
P(Sn < c) = P{(Sn - nmu)/root(nvariance) < (c - nmu)/root(nvariance)} ~~ PHI( (c - nmu)/root(n variance))