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

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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...
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