1. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Suppose X is transformed to another random variable, say h(X).
Then by def,

2. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Suppose X is constant, say b, throughout entire population… b
Then by def,

3. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Suppose X is constant, say b, throughout entire population…
Then…

4. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Multiply X by any constant a… a
Then by def,

5. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Multiply X by any constant a…
Then…
i.e.,…

6. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Multiply X by any constant a…
Add any constant b to X…
Then…
i.e.,…

7. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Multiply X by any constant a…
Add any constant b to X…
Then…
i.e.,…

8. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X

9. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Multiply X by any constant a…
then X is also multiplied by a.

10. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Multiply X by any constant a…
then X is also multiplied by a.
i.e.,…
i.e.,…

11. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Add any constant b to X…
then b is also added to X .

12. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Add any constant b to X…
then b is also added to X .
i.e.,…
i.e.,…

13. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X

14. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
This is the analogue of the “alternate computational formula” for the sample variance s2.

16. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Multiply X by any constant a…
Add any constant b to X…
Then…
i.e.,…

17. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Example: X = Cholesterol level (mg/dL)
random variable X
Discrete
General Properties of “Expectation” of X
Add any constant b to X…
then b is also added to X .
i.e.,…
i.e.,…

18. Random Variable Y = “Sum”
Example: Roll two dice, independently.
Sample space
{(1,1),…,(6,6)}
Discrete
Random Variable Y = “Sum”
0.14
0.16
0.20
0.30
0.15
0.10
0.05
P(Y = y) y
p (y) 2
0.028 3
0.074 4
0.116 5
0.153 6
0.170 7
0.169 8
0.132 9
0.082 10
0.047 11
0.022 12
0.007 1
pmf
pmf
y = 2:12
p = c(.028,.074,
.116,.153,.170, .169,.132,.082, .047,.022,.007)
mu = sum(y*p)
sig.sqd = sum((y-mu)^2*p)
print(c(mu, sig.sqd)) x
p1 (x)
p2 (x) 1
0.20
0.14 2
0.30
0.16 3 4
0.15 5
0.10 6
0.05

19. Example: X = Cholesterol level (mg/dL)
POPULATION
Pop values x
Probabilities
pmf p(x) x1
p(x1) x2
p(x2) x3
p(x3) ⋮
Total 1
Discrete
random variable X
Example: X = Cholesterol level (mg/dL)
General Properties of “Expectation” of X
Theorem
Theorem
If X and Y are independent, then
These formulas apply to both discrete and continuous random variables.
These will be proved in Chapter 8 (Multivariate setting),