advertisement

Probability Distributions for Discrete RV Probability Distributions for Discrete RV Example: Assume we are drawing cards from a 100 well-shuffled cards with replacement. We keep drawing until we get a ♠. Let p = P({♠}), i.e. there are 100 · p ♠’s. Assume the successive drawings are independent and define X = the number of drawings. Probability Distributions for Discrete RV Example: Assume we are drawing cards from a 100 well-shuffled cards with replacement. We keep drawing until we get a ♠. Let p = P({♠}), i.e. there are 100 · p ♠’s. Assume the successive drawings are independent and define X = the number of drawings. If we know that there are 20 ♠’s, i.e. p = 0.2, then what is the probability for us to draw at most 3 times? More than 2 times? Probability Distributions for Discrete RV Example: Assume we are drawing cards from a 100 well-shuffled cards with replacement. We keep drawing until we get a ♠. Let p = P({♠}), i.e. there are 100 · p ♠’s. Assume the successive drawings are independent and define X = the number of drawings. If we know that there are 20 ♠’s, i.e. p = 0.2, then what is the probability for us to draw at most 3 times? More than 2 times? P(X ≤ 3) = p(1)+p(2)+p(3) = 0.2+0.2·0.8+0.2·(0.8)2 = 0.488 Probability Distributions for Discrete RV Example: Assume we are drawing cards from a 100 well-shuffled cards with replacement. We keep drawing until we get a ♠. Let p = P({♠}), i.e. there are 100 · p ♠’s. Assume the successive drawings are independent and define X = the number of drawings. If we know that there are 20 ♠’s, i.e. p = 0.2, then what is the probability for us to draw at most 3 times? More than 2 times? P(X ≤ 3) = p(1)+p(2)+p(3) = 0.2+0.2·0.8+0.2·(0.8)2 = 0.488 P(X > 2) = p(3)+p(4)+p(5)+· · · = 1−p(1)−p(2) = 1−0.2−0.2·0.8 = 0 Probability Distributions for Discrete RV Probability Distributions for Discrete RV Definition The cumulative distribution function (cdf) F (x) of a discrete rv X with pmf p(x) is defined for every number x by X F (x) = P(X ≤ x) = p(y ) y :y ≤x For any number x, F(x) is the probability that the observed value of X will be at most x. Probability Distributions for Discrete RV Definition The cumulative distribution function (cdf) F (x) of a discrete rv X with pmf p(x) is defined for every number x by X F (x) = P(X ≤ x) = p(y ) y :y ≤x For any number x, F(x) is the probability that the observed value of X will be at most x. F (x) = P(X ≤ x) = P(X is less than or equal to x) p(x) = P(X = x) = P(X is exactly equal to x) Probability Distributions for Discrete RV Probability Distributions for Discrete RV Example 3.10 (continued): 0 0.4 F (y ) = 0.7 0.9 1 if if if if if y <1 1≤y <2 2≤y <3 3≤y <4 y ≥2 Probability Distributions for Discrete RV Example 3.10 (continued): 0 0.4 F (y ) = 0.7 0.9 1 if if if if if y <1 1≤y <2 2≤y <3 3≤y <4 y ≥2 Probability Distributions for Discrete RV Probability Distributions for Discrete RV Example: Assume we are drawing cards from a 100 well-shuffled cards with replacement. We keep drawing until we get a ♠. Let α = P({♠}), i.e. there are 100 · α ♠’s. Assume the successive drawings are independent and define X = the number of drawings. The pmf would be ( (1 − α)x−1 · α x = 1, 2, 3, . . . p(x) = 0 otherwise Probability Distributions for Discrete RV Example: Assume we are drawing cards from a 100 well-shuffled cards with replacement. We keep drawing until we get a ♠. Let α = P({♠}), i.e. there are 100 · α ♠’s. Assume the successive drawings are independent and define X = the number of drawings. The pmf would be ( (1 − α)x−1 · α x = 1, 2, 3, . . . p(x) = 0 otherwise Then for any positive interger x, we have F (x) = X y ≤x p(y ) = x x−1 X X (1 − α)(y −1) · α = α (1 − α)y y =1 ( 1 − (1 − α)x = 0 y =0 x ≥1 x <1 Probability Distributions for Discrete RV Probability Distributions for Discrete RV Example: Assume we are drawing cards from a 100 well-shuffled cards with replacement. We keep drawing until we get a ♠. Let α = P({♠}), i.e. there are 100 · α ♠’s. Assume the successive drawings are independent and define X = the number of drawings. The pmf would be Probability Distributions for Discrete RV Probability Distributions for Discrete RV pmf =⇒ cdf: F (x) = P(X ≤ x) = X y :y ≤x p(y ) Probability Distributions for Discrete RV pmf =⇒ cdf: F (x) = P(X ≤ x) = X y :y ≤x It is also possible cdf =⇒ pmf: p(y ) Probability Distributions for Discrete RV pmf =⇒ cdf: F (x) = P(X ≤ x) = X p(y ) y :y ≤x It is also possible cdf =⇒ pmf: p(x) = F (x) − F (x−) where “x−” represents the largest possible X value that is strictly less than x. Probability Distributions for Discrete RV Probability Distributions for Discrete RV Proposition For any two numbers a and b with a ≤ b, P(a ≤ X ≤ b) = F (b) − F (a−) where “a−” represents the largest possible X value that is strictly less than a. In particular, if the only possible values are integers and if a and b are integers, then P(a ≤ X ≤ b) = P(X = a or a + 1 or . . . or b) = F (b) − F (a − 1) Taking a = b yields P(X = a) = F (a) − F (a − 1) in this case. Probability Distributions for Discrete RV Probability Distributions for Discrete RV Example (Problem 23): A consumer organization that evaluates new automobiles customarily reports the number of major defects in each car examined. Let X denote the number of major defects in a randomly selected car of a certain type. The cdf of X is as follows: 0 x <0 0.06 0 ≤ x < 1 0.19 1 ≤ x < 2 0.39 2 ≤ x < 3 F (x) = 0.67 3 ≤ x < 4 0.92 4 ≤ x < 5 0.97 5 ≤ x < 6 1 x ≤6 Calculate the following probabilities directly from the cdf: (a)p(2), (b)P(X > 3) and (c)P(2 ≤ X < 5).