1. MATH 3033 based on Dekking et al
MATH 3033 based on Dekking et al. A Modern Introduction to Probability and Statistics Slides by Yu Liang Instructor Longin Jan Latecki
Chapter 7: Expectation and variance

2. The expectation of a discrete random variable X taking the values a1, a2, and with probability mass function p is the number:
We also call E[X] the expected value or mean of X. Since the expectation is
determined by the probability distribution of X only, we also speak of the
expectation or mean of the distribution.
Expected values
of discrete random variable

3. The expectation of a continuous random variable X with probability density function f is the number
We also call E[X] the expected value or mean of X. Note that E[X] is indeed
the center of gravity of the mass distribution described by the function f:
Expected values
of continuous random variable

4. The EXPECTATION of a GEOMETRIC DISTRIBUTION
The EXPECTATION of a GEOMETRIC DISTRIBUTION. Let X have a geometric distribution with parameter p; then
The EXPECTATION of an EXPONENTIAL DISTRIBUTION. Let X have an exponential distribution with parameter λ; then
The EXPECTATION of a GEOMETRIC DISTRIBUTION. Let X have a geometric distribution with parameter p; then

5. The CHANGE-OF-VARIABLE FORMULA
The CHANGE-OF-VARIABLE FORMULA. Let X be a random variable, and let g : R → R be a function.
If X is discrete, taking the values a1, a2, , then
If X is continuous, with probability density function f, then

6. The variance Var(X) of a random variable X is the number
Standard deviation:
Variance of a normal distribution. Let X be an N(μ, σ2) distributed random variable. Then

7. An alternative expression for the variance. For any random variable X,
is called the second moment of X. We can derive this equation from:
Expectation and variance under change of units. For any random variable X and any real numbers r and s,
and