Topic 3 lecture note.docx - TEST 3 PROBABILITY DISTRIBUTION I Discrete distribution(Uniform distribution Discrete random variable A random variable that

Topic 3 lecture note.docx - TEST 3 PROBABILITY DISTRIBUTION...

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TEST 3 - PROBABILITY DISTRIBUTION I. Discrete distribution (Uniform distribution) - Discrete random variable : A random variable that can assume only certain, clearly separated values. Usually used for counting something - Example : The number of heads appearing when a coin is tossed three times. Example: When roll a die 1 2 3 4 5 6 P (x = x) 1/6 1/6 1/6 1/6 1/6 1/6 E(X) = 1 ( 1 6 ) + 2( 1 6 ¿ + 3 1 6 ¿ ) + 4 ( 1 6 ¿ + 5 ( 1 6 ¿ + 6 ( 1 6 ¿ = 7 2 Or a c 3.5 d b 0.5 1 2 3 4 5 6 6.5 p/s: a is 0.5 less and b is 0.5 more E(X) = b + a 2 = 0.5 + 6.5 2 = 7 2 V(X) = ( b + a ) 2 12 = 6.5 + 0.5 12 = 7 12 P (c ≤ x ≤) = d c b a = 5 6 II. Bernoulli distribution - Running only 1 experiment that either lead to and success or failure -> Example : What is the probability of getting an even number when a fair die is thrown once ? - Bernoulli trial : Yes (1) and No (0) - p: success - q : Failure or P(1-p) Example: Flipping coin X 0 1 P (x = x) 0.5 0.5 X ~ Ber (0.5) E(X) = P = 0.5 V(X) = E(X 2 ) - E(X) 2 = p q = success x failure = 0.5 x 0.5 = 0.25 f ( x ) = 1 a b
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III. Geometric distribution - The geometric distribution represents the number of failures before you get a success in Bernoulli trials . -> Example: when buying lottery tickets, the first winning ticket is obtained on the fourth purchase -> Example: what is the probability you meet an independent voter on your third try ? - X ~ Geo (0.5) - P (X=X) = p q x-1 X 1 2 3 …… P (X = X) Pq 0 0.5 Pq 1 0.5 x 0.5 Pq 2 0.5 x 0.5 2 Pq P (X = 1) = 0.5 P (X =2) = 0.25…. E(X) = 1 P = 1 0.5 = 2 V(X) = q P 2 = 0.5 0.5 x 0.5 = 2 IV. Binomial distribution - A binomial distribution is the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. -> Example: Hospital records show that of patients suffering from a certain disease, 75% die because of it. What is the probability that of 6 randomly selected patients, 4 will recover ? Example: If 5 people come to the room, what’s the chance of 2 of them are International students => n = 5, P = 0.5 X ~ Bin (0.5, 5) - Probability of International student is 0.5 - Probability of not international student is 0.5 => P (X = 2) = P (X 1 = 0) P (X 2 = 0) P(X 3 =0) P (X 4 = 1) P (X 5 = 1) => P (X=2) = 5C 2 x P 2 q 3 = 0.3125 Note that it’s any success, so it doesn’t matter if X 1 = 0 or X 1 =1, just have to keep the success rate at 2 Example: Flipping a coin 2 times Table II Or +) Step 1: Create the probability table for flipping coin 1 st one and 2 nd one X 1 0 1 0.5 0.5
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