Chapter 4. Chapter 4 sections

Transcription

1 Chapter 4 sections 4.1 Expectation 4.2 Properties of Expectations 4.3 Variance 4.4 Moments 4.5 The Mean and the Median 4.6 Covariance and Correlation 4.7 Conditional Expectation SKIP: 4.8 Utility Expectation 1 / 18

2 Summarizing distributions 4.1 Expectation The distribution of X contains everything there is to know about the probabilistic properties of X. However, sometimes we want to summarize the distribution of X in one or a few numbers e.g. to more easily compare two or more distributions. Examples of descriptive quantities: Mean ( = Expectation ) Center of mass - weighted average Median, Moments Variance, Interquartile Range (IQR), Covariance, Correlation Expectation 2 / 18

3 4.1 Expectation Definition of Expectation µ = E(X) Def: Mean aka. Expected value Let X be a random variable with p(d)f f (x). The mean, or expected value of X, denoted E(X), is defined as follows X discrete: E(X) = xf (x) All x assuming the sum exists. X continuous: E(X) = assuming the integral exists. xf (x) dx If the sum or integral does not exists we say that the expected value does not exist. The mean is often denoted with µ. Expectation 3 / 18

4 Examples Chapter Expectation Recall the distribution of Y = the number of heads in 3 tosses (coin toss example from Lecture 4) y then f Y (y) 1 8 E(Y ) = = 12 8 = 3 2 = 1.5 Find E(X) where X Binom(n, p). The pf of X is ( ) n f (x) = p x (1 p) n x for x = 0, 1,..., n x Find E(X) where X Uniform(a, b). The pdf of X is f (x) = 1 b a for a x b Expectation 4 / 18

5 4.1 Expectation Expectation of g(x) Theorem Let X be a random variable with p(d)f f (x) and g(x) be a real-valued function. Then X discrete: E (g(x)) = All x g(x)f (x) X continuous: E (g(x)) = g(x)f (x) dx Example: Find E(X 2 ) where X Uniform(a, b). Expectation 5 / 18

6 Expectation of g(x, Y ) Chapter Expectation Theorem Let X and Y be random variables with joint p(d)f f (x, y) and let g(x, y) be a real-valued function. Then X and Y discrete: X and Y continuous: E (g(x, Y )) = E (g(x, Y )) = All x,y g(x, y)f (x, y) g(x, y)f (x, y) dx dy Example: Find E ( ) X+Y 2 where X and Y are independent and X Uniform(a, b) and Y Uniform(c, d). Expectation 6 / 18

7 Properties of Expectation 4.2 Properties of Expectations Theorems 4.2.1, and 4.2.6: E(aX + b) = ae(x) + b for constants a and b. Let X 1,..., X n be n random variables, all with finite expectations E(X i ), then ( n ) n E X i = E(X i ) Corollary: E (a 1 X a n X n + b) = a 1 E(X 1 ) + + a n E(X n ) + b for constants b, a 1,..., a n. Let X 1,..., X n be n independent random variables, all with finite expectations E(X i ), then ( n ) n E X i = E(X i ) CAREFUL!!! In general E(g(X)) g(e(x)). For example: E(X 2 ) [E(X)] 2 Expectation 7 / 18

8 Examples Chapter Properties of Expectations If X 1, X 2,..., X n are i.i.d. Bernoulli(p) random variables then Y = n X i Binomial(n, p). E(X i ) = 0 (1 p) + 1 p = p for i = 1,..., n ( n ) n n E(Y ) = E X i = E(X i ) = p = np Note: i.i.d. stands for independent and identically distributed Expectation 8 / 18

9 4.3 Variance Definition of Variance σ 2 = Var(X) Def: Variance Let X be a random variable (discrete or continuous) with a finite mean µ = E(X). The Variance of X is defined as Var(X) = E ((X µ) 2) The standard deviation of X is defined as Var(X) We often use σ 2 for variance and σ for standard deviation. Theorem Another way of calculating variance For any random variable X Var(X) = E(X 2 ) [E(X)] 2 Expectation 9 / 18

10 4.3 Variance Examples - calculating the variance Recall the distribution of Y = the number of heads in 3 tosses (coin toss example from Lecture 4) y f Y (y) We already found that µ = E(Y ) = 1.5. Then Var(Y ) = (0 1.5) (1 1.5)2 3 8 = (2 1.5) (3 1.5)2 1 8 Find Var(X) where X Uniform(a, b) Expectation 10 / 18

11 Properties of the Variance 4.3 Variance Theorems 4.3.2, 4.3.3, and Var(X) 0 for any random variable X. Var(X) = 0 if and only if X is a constant, i.e. P(X = c) = 1 for some constant c. Var(aX + b) = a 2 Var(X) If X 1,..., X n are independent we have ( n ) n Var X i = Var(X i ) Expectation 11 / 18

12 Examples Chapter Variance If X 1, X 2,..., X n are i.i.d. Bernoulli(p) random variables then Y = n X i Binomial(n, p). E(X i ) = p for i = 1,..., n E(Xi 2 ) = 0 2 (1 p) p = p for i = 1,..., n Var(X i ) = E(Xi 2 ) [E(X i )] 2 = p p 2 = p(1 p) Therefore, we have ( n ) n n Var(Y ) = Var X i = Var(X i ) = p(1 p) = np(1 p) Expectation 12 / 18

13 4.4 Moments Moments and Central moments Def: Moments Let X be a random variable and k be a positive integer. The expectation E(X k ) is called the k th moment of X Let E(X) = µ. The expectation E ( (X µ) k) is called the k th central moment of X The first moment is the mean: µ = E(X 1 ) The first central moment is zero: E(X µ) = E(X) E(X) = 0 The second central moment is the variance: σ 2 = E ( (X µ) 2) Expectation 13 / 18

14 4.4 Moments Moments and Central moments Symmetric distribution: If the p(d)f f (x) is symmetric with respect to a point x 0, i.e. f (x 0 + δ) = f (x 0 δ) for all δ It the mean of a symmetric distribution exists, then it is the point of symmetry. If the distribution of X is symmetric w.r.t. its mean µ then E ( (X µ) k) = 0 for k odd (if the central moment exists) Skewness: E ( (X µ) 3) /σ 3 Expectation 14 / 18

15 Moment generating function 4.4 Moments Def: Moment Generating Function Let X be a random variable. The function ( ) ψ(t) = E e tx t R is called the moment generating function (m.g.f.) of X Theorem Let X be a random variables whose m.g.f. ψ(t) is finite for t in an open interval around zero. Then the nth moment of X is finite, for n = 1, 2,..., and E(X n ) = d n dt n ψ(t) t=0 Expectation 15 / 18

16 Example Chapter Moments Let X Gamma(n, β). Then X has the pdf f (x) = 1 (n 1)!β n x n 1 e x/β for x > 0 Find the m.g.f. of X and use it to find the mean and the variance of X. Expectation 16 / 18

17 Properties of m.g.f. Chapter Moments Theorems and 4.4.4: ψ ax+b (t) = e bt ψ X (at) Let Y = n X i where X 1,..., X n are independent random variables with m.g.f. ψ i (t) for i = 1,..., n Then ψ Y (t) = n ψ i (t) Theorem 4.4.5: Uniqueness of the m.g.f. Let X and Y be two random variables with m.g.f. s ψ X (t) and ψ Y (t). If the m.g.f. s are finite and ψ X (t) = ψ Y (t) for all values of t in an open interval around zero, then X and Y have the same distribution. Expectation 17 / 18

18 4.4 Moments Example Let X N(µ, σ 2 ). X has the pdf f (x) = 1 ) ( σ 2π exp (x µ)2 2σ 2 and the m.g.f. for the normal distribution is Show that ψ(t) is the m.g.f. of X. ψ(t) = exp (µt + t2 σ 2 ) 2 Expectation 18 / 18