1. Random Variable X, with pmf p(x) or pdf f(x)
POPULATION
Random Variable X, with pmf p(x) or pdf f(x)
Recall…
REVIEW POWERPOINT SECTION “ ” FOR BASIC PROPERTIES OF EXPECTED VALUE
PARAMETERS
“population characteristics”
Mean:
Consider a transformation to a new random variable g(X). Then…
Example:

2. Random Variable X, with pmf p(x) or pdf f(x)
POPULATION
Random Variable X, with pmf p(x) or pdf f(x)
Recall…
REVIEW POWERPOINT SECTION “ ” FOR BASIC PROPERTIES OF EXPECTED VALUE
PARAMETERS
“population characteristics”
Mean:
Variance:
Proof: See PowerPoint section , slides 56, 57 for discrete X.

3. Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
POPULATION
Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
Let g(X, Y) be a real-valued function of X and Y, i.e.,
Then….

4. Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
POPULATION
Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
Let g(X, Y) be a real-valued function of X and Y, i.e.,
Then….
Example:
Find the mean distance from any random point inside a circle of fixed radius a > 0 to its center.
Assume that points (X, Y) are uniformly distributed. ..

5. Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
POPULATION
Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
Let g(X, Y) be a real-valued function of X and Y, i.e.,
Then….
Example:
Find the mean distance from any random point inside a circle of fixed radius a > 0 to its center.
Assume that points (X, Y) are uniformly distributed. ..

6. Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
POPULATION
Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
Let g(X, Y) be a real-valued function of X and Y, i.e.,
Then….
Example:
Find the mean distance from any random point inside a circle of fixed radius a > 0 to its center.
Assume that points (X, Y) are uniformly distributed. ..

7. Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
POPULATION
Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
Let g(X, Y) be a real-valued function of X and Y, i.e.,
Then….
Example:
Find the mean distance from any random point inside a circle of fixed radius a > 0 to its center.
Assume that points (X, Y) are uniformly distributed. ..

8. # This segment constructs a graph of the unit circle.
plot.new()
f1 = function(x)(sqrt(1-x^2))
f2 = function(x)(-1*f1(x))
plot(f1, xlim=range(-1,1), ylim=range(-1,1))
curve(f2, add = T)
# This segment simulates N uniformly distributed points in the
# unit disk, and computes their average distance to its center.
N = 1000
counter = 0
dist = NULL
while(counter < N) {
x = runif(1, -1, 1)
y = runif(1, -1, 1)
z = sqrt(x^2 + y^2)
if (z < 1) {
dist = c(dist, z)
counter = counter + 1
points(x, y, pch=19, cex=0.5)} }
mean(dist)
[1]

9. Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
POPULATION
Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
Let g(X, Y) be a real-valued function of X and Y, i.e.,
Then….
Extension to multiple random variables X1, X2, X3,…, Xn

10. Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
POPULATION
Random Variables X and Y, with joint pmf p(x, y) or pdf f(x, y)
If X and Y are independent, then…
Proof:
Exercise
(next section…)