Download Section 4.1: Mean of a Random Variable

STAT 381
Chapter 4
Notes
Section 4.1: Mean of a Random Variable
Definitions
1. Mean / Expected Value of X
• Let X be a random variable with probability distribution f (x). The mean or expected
value of X is
(P
xf (x),
if X is discrete
µX = E(X) = R 1x
xf (x)dx, if X is continuous.
1
• This is a long-run average.
• It describes where the probability distribution is centered.
2. Expected Value of g(X)
• Let X be a random variable with probability distribution f (x). The expected value
of the random variable g(X) is
(P
g(x)f (x),
if X is discrete
µg(X) = E [g(X)] = R 1x
g(x)f (x)dx, if X is continuous.
1
• g(X) is any function of X. Examples: g(X) = X 2 ; g(X) = (3X + 1)3 etc.
3. Expected Value of the Joint Distribution (Special Cases)
• There is a formula for the expected value of the joint distribution for X and Y .
However, we are mostly concerned with the special cases listed below.
• Let X and Y be random variables with joint probability distribution f (x, y). The
mean or expected value of the random variable g(X, Y ) for two special cases are the
following. Let g(x) be the marginal distribution of X. Let h(y) be the marginal
distribution of Y .
Case 1: If g(X, Y ) = X, then the marginal expected value with respect to X is
(P
xg(x),
if X is discrete
µX = E(X) = R 1x
xg(x)dx, if X is continuous.
1
Case 2: If g(X, Y ) = Y , then the marginal expected value with respect to Y is
(P
yh(y),
if Y is discrete
µY = E(Y ) = R 1y
yh(y)dy, if Y is continuous.
1
Section 4.2: Variance and Covariance
Definitions
1. Variance of X
• Let X be a random variable with probability distribution f (x) and mean µ. The
variance of X is
(P
⇥
⇤
(x µ)2 f (x),
if X is discrete
2
= V(X) = E (X µ)2 = R 1x
(x µ)2 f (x)dx, if X is continuous.
1
• The standard deviation of X is
=+
p
2.
• Variance helps to describe the shape of the distribution.
• If variability from the mean is small, then
mean is large, then 2 is large.
2
is small. If the variability from the
Theorem 1. The variance of a random variable X is
2
= V(X) = E(X 2 )
µ2 = E(X 2 )
[E(X)]2 .
2. Variance of g(X)
• Let X be a random variable with probability distribution f (x). The variance of the
random variable g(X) is
(P ⇥
⇤2
n⇥
o
⇤
µg(X) f (x),
if X is discrete
2
2
x g(x)
⇤2
g(X) µg(X)
= R1 ⇥
g(X) = E
g(x) µg(X) f (x)dx, if X is continuous.
1
• g(X) is any function of X. Examples: g(X) = X 2 ; g(X) = 2X 2 + X.
2
Section 4.3: Means and Variances of Linear Combinations of Random Variables
Rules for Expectations
Suppose X and Y are random variables, and a and b are constant numbers. Suppose g and h
are functions.
1. E(X + Y ) = E(X) + E(Y )
2. E(X
Y ) = E(X)
E(Y )
3. E(aX) = a E(X)
4. E(b) = b
5. E(aX + b) = a E(X) + b
6. E [g(X) ± h(X)] = E [g(X)] ± E [h(X)]
7. If X and Y are independent, then E(XY ) = E(X) E(Y ).
Rules for Variances
Suppose X, Y and Z are independent random variables, and a, b and c are constant numbers.
1. V(X + Y ) = V(X) + V(Y )
2. V(X
Y ) = V(X) + V(Y )
3. V(aX) = a2 V(X)
4. V(b) = 0
5. V(aX + b) = a2 V(X)
6. V(aX + bY + cZ) = a2 V(X) + b2 V(Y ) + c2 V(Z)
7. Suppose Z = 3X + Y and X and Y are independent. Then V(Z) = V(3X + Y ) =
9 V(X) + V(Y ).
3