1. Introduction to Probability & Statistics Joint Expectations

2. Properties of Expectations
Recall that:
1. E[c] = c
2. E[aX + b] = aE[X] + b
3. 2(ax + b) = a22
4. E[g(x)] = g x dF ( ) 

3. Joint Expectation Let Z = X + Y E[Z] = E[X] + E[Y]  ( ) cov( , z x y2 ( )
cov( , z x y  

4. Derivation   E Z zf x y dxdy [ ] ( , )    ( ) , x y f dxdy
Let Z = X + Y E Z zf x y
dxdy XY [ ] ( , )      ( ) , x y f
dxdy XY

5. Derivation E[Z]      E Z zf x y dxdy [ ] ( , )    ( ) , x y f
Let Z = X + Y E Z zf x y
dxdy XY [ ] ( , )      ( ) , x y f
dxdy XY    y XY x xf
dxdy yf ( , )    x f y
dydx
dxdy XY ( , )    xf x dx yf y dy X Y ( )
= E[X] + E[Y]

6. Derivation of 2(z)   ( ) , [ ] z f x y dxdy E Z  
Miracle 13 occurs
= 2(x) + 2(y)

7. In General
In general, for Z = the sum of n independent variates, Z = X1 + X2 + X Xn E Z X i n [ ]   1  2 ( ) z x

8. Class Problem
Suppose n customers enter a store. The ith customer spends some Xi amount. Past data indicates that Xi is a uniform random variable with mean $100 and standard deviation $5. Determine the mean and variance for the total revenue for 5 customers.

9. Class Problem Total Revenue is given by TR = X1 + X2 + X3 + X4 + X5
Using the property E[Z] = E[X] + E[Y],
E[TR] = E[X1] + E[X2] + E[X3] + E[X4] + E[X5] =
= $500

10. Class Problem TR = X1 + X2 + X3 + X4 + X5
Use the property s2(Z) = s2(X) + s2(Y).
Assuming Xi, i = 1, 2, 3, 4, 5 all independent
s2(TR) = s2(X1) + s2(X2) + s2(X3) + s2(X4) + s2(X5) =
= 125

11. Class Problem TR = X1 + X2 + X3 + X4 + X5
Xi , independent with mean 100 and standard deviation 5
E[TR] = $500
s2(TR) = 125