#### Transcript C7_Math3033

MATH 3033 based on Dekking et al. A Modern Introduction to Probability and Statistics. 2007 Slides by Yu Liang Instructor Longin Jan Latecki Chapter 7: Expectation and variance The expectation of a discrete random variable X taking the values a1, a2, . . . and with probability mass function p is the number: E[ X ] ai P(X ai ) ai p(ai ) i i 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 The expectation of a continuous random variable X with probability density function f is the number E[ X ] xf ( x)dx 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: E[ X ] xf ( x)dx xf ( x)dx - f ( x)dx - Expected values of continuous random variable The EXPECTATION of a GEOMETRIC DISTRIBUTION. Let X have a geometric distribution with parameter p; then E[ X ] kp(1 p)k 1 k 1 1 p The EXPECTATION of an EXPONENTIAL DISTRIBUTION. Let X have an exponential distribution with parameter λ; then E[ X ] xe dx x 0 1 The EXPECTATION of a GEOMETRIC DISTRIBUTION. Let X have a geometric distribution with parameter p; then 1 X ( 1 E[ X ] x e 2 2 )2 dx 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 E[ g ( X )] g (ai )P(X ai ) i If X is continuous, with probability density function f, then E[ g ( X )] g ( x ) f ( x ) dx The variance Var(X) of a random variable X is the number Var ( X ) E[( X E[ X ])2 ] Standard deviation: Var ( X ) Variance of a normal distribution. Let X be an N(μ, σ2) distributed random variable. Then 1 x ( 1 2 Var ( X ) ( x ) e 2 2 )2 dx 2 An alternative expression for the variance. For any random variable X, Var ( X ) E[ X 2 ] E[ X ]2 E[ X 2 ] is called the second moment of X. We can derive this equation from: Var ( X ) E[( X E[ X ]2 ] ( x E[ X ]2 f ( x )dx ( x 2 2 xE[ X ] ( E[ X ])2 ) f ( x )dx x 2 f ( x )dx 2 E[ X ] xf ( x )dx ( E[ X ])2 f ( x )dx E[ X 2 ] 2( E[ X ])2 ( E[ X ])2 ( E[ X ]) ( E[ X ]) 2 2 Expectation and variance under change of units. For any random variable X and any real numbers r and s, E[rX s] rE[ X ] s and Var (rX s) r 2Var ( X )