Section 5 – Expectation and Other Distribution
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Transcript Section 5 – Expectation and Other Distribution
Section 5 – Expectation and
Other Distribution Parameters
Expected Value (mean)
• As the number of trials increases, the average
outcome will tend towards E(X): the mean
• Expectation: E[X] X
– Discrete
E[X] x p(x) x1 p(x1) x2 p(x2 ) ...
– Continuous
E[X]
x f (x)dx
Expectation of h(x)
• Discrete
E[X] x p(x) x1 p(x1) x2 p(x2 ) ...
E[h(X)] h(x) p(x) h(x1) p(x1) h(x 2 ) p(x 2 ) ...
x
• Continuous
E[X]
x f (x)dx
E[h(X)]
h(x) f (x)dx
Moments of a Random Variable
• n: positive integer
• n-th moment of X:
E[X n ]
– So h(x) = X^n
– Use E[h(X)] formula in previous slide
• n-th central moment of X (about the mean):
E[(X ) n ]
– Not as important to know
Variance of X
• Notation
• Definition:
Var[X] V[X]
Var[X]
E[(X X ) ]
2
E[X 2 ] (E[X])2
E[X ]
2
2
X
2
X
2
Important Terminology
• Standard Deviation of X: X 2X Var[X]
• Coefficient of variation:
X
X
– Trap: “Coefficient of variation” uses standard
deviation not variance.
Moment Generating Function (MGF)
• Moment generating function of a random
tX
variable X:
MX (t) E[e ]
– Discrete:
MX (t) etx p(x)
– Continuous: M X (t) e tx f (x)dx
Properties of MGF’s
M X (0) 1
M'X (0) E[X ]
M''X (0) E[X ]
2
M (n )X (0) E[X n ]
d2
ln[M X (t)] |t 0 Var[X]
2
dt
Two Ways to Find Moments
1. E[e^(tx)]
2. Derivatives of the MGF
Characteristics of a Distribution
• Percentile: value of X, c, such that p% falls to the left of c
– Median: p = .5, the 50th percentile of the distribution (set CDF integral
=.5)
• What if (in a discrete distribution) the median is between two numbers? Then
technically any number between the two. We typically just take the average
of the two though
• Mode: most common value of x
– PMF p(x) or PDF f(x) is maximized at the mode
• Skewness: positive is skewed right / negative is skewed left
E[( X ) 3 ]
3
– I’ve never seen the interpretation on test questions, but the formula
might be covered to test central moments and variance at the same
time
Expectation & Variance of Functions
• Expectation: constant terms out, coefficients
out
E[a1h1 ( X ) a2 h2 ( X ) b] a1E[h1 ( X )] a2 E[h2 ( X )] b
E[aX b] aE[ X ] b
• Variance: constant terms gone, coefficients
out as squares
Var[aX b] a2Var[X]
Mixture of Distributions
• Collection of RV’s X1, X2, …, Xk
– With probability functions f1(x), f2(x), …, fk(x)
– These functions have weights (alpha) that sum to 1
• In a “mixture of distribution” these
distributions are mixed together by their
weights
– It’s a weighted average of the other distributions
f (x) 1 f1(x) 2 f 2 (x) ... k f k (x)
Parameters of Mixtures of Distributions
E[X n ] 1 E[X1n ] 2 E[X 2n ] ... k E[X kn ]
M X (t) 1 M X1 (t) 2 M X 2 (t) ... k M X k (t)
• Trap: The Variance is NOT a weighted average
of the variances
• You need to find E[X^2]-(E[X)])^2 by finding
each term for the mixture separately