Chen iecuhk npx n n n1 engg2430c lecture 5 1 2 n1 1 1

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: variable X= 1, the coin gives HEAD; 0, the coin gives TAIL. E [X ] = p × 1 + (1 − p ) × 0 = p Calculate the expectation of a Geometric random variable with probability of getting HEAD is p = 1 2 The expectation of X is given by ∞ E [X ] = n=1 M. Chen (IE@CUHK) ∞ ∑ npX (n) = ∑ n × n=1 ENGG2430C lecture 5 1 2 n−1 1− 1 2 = 2. 16 / 21 Variance: Degree of Randomness of the Outcome* Definition: Variance of a r.v. X : Var(X ) E (X − E [X ])2 How the probability mass concentrates around E [X ]. Degree of randomness of outcomes. Standard deviation of a r.v. X : σ (X ) = Var(X ) Prove that Var(X ) = E X 2 − E 2 [X ] M. Chen (IE@CUHK) ENGG2430C lecture 5 17 / 21 Examples Consider a random variable X uniformly distributed in {a, a + 1, · · · , b }, where a and b are two integers (b ≥ a) pX ( x ) = P ( X = x ) = Compute E [X ]. E [X ] = 0, if x = a, a + 1, · · · , b ; otherwise. a +b 2 Compute Var (X ). Var (X ) = M. Chen (IE@CUHK) 1 b −a+1 , (b −a+1)2 −1 12 ENGG2430C lecture 5 18 / 21 Examples X : the interval between two consecutive buses arriving at a station: Case 1: X uniformly distributes in {3, 4, 5} minutes. 2 Var(X ) = . 3 Case 2: X uniformly distributes in {1, 2, 3, 4, 5, 6, 7} minutes. E [X ] = 4, E [X ] = 4, Var(X ) = 28 = 4. 7 Question: What does Var(X ) = 0 imply? How about Var(X ) → ∞? M. Chen (IE@CUHK) ENGG2430C lecture 5 19 / 21 Examples Let X be a Bernoulli random variable. pX (x ) = P (X = x ) = p, if x = 1; 1 − p , if x = 0; E [X ] = p . Var(X ) =?p − p 2 = p (1 − p ) Let X be a Geometric random variable. pX (k ) = P (X = k ) = (1 − p )k −1 p , if k = 1, 2, · · · ; 0, otherwise; 1 E [X ] = p . Var(X ) =?(Hint: let S = E X 2 . Compute S − (1 − p )S ; you may find 1 E [X ] handy in simplifying the calculation.) p12 − p M. Chen (IE@CUHK) ENGG2430C lecture 5 20 / 21 Thank You Reading: Ch. 2.1-2.4 of the textbook. Next lecture: Examples of Random Variables, Joint PMF, and Independence M. Chen (IE@CUHK) ENGG2430C lecture 5 21 / 21...
View Full Document

Ask a homework question - tutors are online