Example_class_4_handout_solution

Example_class_4_handout_solution - THE UNIVERSITY OF HONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 4 Review Abbreviated event notation Usually we abbreviate the description of the event or the set { ω : X ( ω ) ≤ x } to simply as { X ≤ x } so that we can also abbreviate the probability notation P ( { ω : X ( ω ) ≤ x } ) as P ( { X ≤ x } ) or even simpler as P ( X ≤ x ). This is primarily because the two probability measures P X and P are consistent due to the property of the random variable X . Definition of CDF F X ( x ) = P ( X ≤ x ) = P ( { ω : X ( ω ) ≤ x } ), where x ∈ (-∞ , + ∞ ) Properties of CDF (a) 0 ≤ F X ( x ) ≤ 1 (b) If x 1 < x 2 , then F X ( x 1 ) ≤ F X ( x 2 ). (c) lim x → + ∞ F X ( x ) = F X (+ ∞ ) = 1 (d) lim x →-∞ F X ( x ) = F X (-∞ ) = 0 (e) lim x → a + F X ( x ) = F X ( a + ) = F X ( a ), where a + = lim < → ( a + ) (f) P ( a < X ≤ b ) = F X ( b )- F X ( a ) (g) P ( X > a ) = 1- F X ( a ) (h) P ( X < b ) = F X ( b- ) , where b- = lim < → ( b- ) Definition of PMF p X ( x ) = P ( X = x ) = P ( { ω : X ( ω ) = x } ), where x ∈ (-∞ , + ∞ ) 1 Properties of PMF The state space of a discrete random variable X must be countable. Denote the state space of a discrete r.v. X by S = { x 1 ,x 2 ,x 3 ,... } . (a) 0 ≤ p X ( x k ) ≤ 1, k = 1 , 2 ,... (b) If x 6 = x k for all k = 1 , 2 ,... , then p X ( x ) = 0 (c) X k p X ( x k ) = 1 (d) F X ( x ) = ∑ x k ≤ x p X ( x k ) Remark. P ( E ) = 0 ; E = φ Remark. If X is a continuous random variable, then P ( X = x ) = 0. Expectation E ( X ) E ( g ( X )) X is discrete X x ∈ X (Ω) xp ( x ) X x ∈ X (Ω) g ( x ) p ( x ) X is continuous ˆ X (Ω) xf ( x ) dx ˆ X (Ω) g ( x ) f ( x ) dx Mean and Variance Mean of X = E ( X ) Variance of X = V ar ( X ) = E (( X- E ( X )) 2 ) = E ( X 2 )- E ( X ) 2 . The mean is usually denoted by μ and the variance is usually denoted by σ 2 . Moment and moment generating function Moment. Let r be a positive integer. The r th moment of X is E ( X r ). The r th moment of X about b is E (( X- b ) 2 ). Moment generating function. Let X be a random variable. The moment generating function of X is defined as M X ( t ) = E ( e tX ) if it exists. The domain of M X ( t ) is all real number t such that the expectation E ( e tX ) is finite. Also, M ( r ) X (0) = ∂ r M X ( t ) ∂t r t =0 = E ( X r ) . 2 Remark. Moment generating function uniquely characterizes the distribution....
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Example_class_4_handout_solution - THE UNIVERSITY OF HONG...

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