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Lesson #12 Discrete Probability Distributions

Lesson #12 Discrete Probability Distributions. Y = # people in ER in an hour. Flip 2 coins, X = # heads. y P(y). x P(x). p 0 p 1 p 2 p 3 p 4 . 0 1 2 3 4 . 0 1 2. .25 .50 .25. x = 0, 1, 2. For discrete distributions, f(x) is called the probability function.

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Lesson #12 Discrete Probability Distributions

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  1. Lesson #12 Discrete Probability Distributions

  2. Y = # people in ER in an hour Flip 2 coins, X = # heads yP(y) xP(x) p0 p1 p2 p3 p4  0 1 2 3 4  0 1 2 .25 .50 .25

  3. x = 0, 1, 2 For discrete distributions, f(x) is called the probabilityfunction. f(c) = P(X = c) Flip 2 coins, X = # heads

  4. For discrete distributions, F(x) is called the cumulativedistributionfunction, or cdf. F(c) = P(X < c) Flip 2 coins, X = # heads xf(x)F(x) 0 1 2 .25 .50 .25 .25 .75 1.00

  5. x = 0, 1, 2, 3 X = # of side effects from a certain medication P(X = 0) = f(0) = .375 P(X = 1) = f(1) = .325

  6. xf(x)F(x) 0 1 2 3 .375 .325 .225 .075 1.000 .375 .700 .925 1.000 P(at least two side effects) = P(X > 2) = 1 – P(X < 1) = 1 – F(1) = 1 - .700 = .300 P(no more than two side effects) = P(X < 2) = F(2) = .925

  7. Mean of a discrete distribution m = E(X) = x • f(x) Variance of a discrete distribution s2 = Var(X) = (x - m)2 • f(x) Standard deviation,

  8. xf(x)F(x) 0 1 2 .25 .50 .25 .25 .75 1.00 0 1 2 m = (0)(.25) + (1)(.50) + (2)(.25) = 1 s2 = (0-1)2(.25) + (1-1)2(.50) + (2-1)2(.25) = .5 xfrfcfrcf 0 1 2 29 45 26 100 .29 .45 .26 29 74 100 .29 .74 1.00

  9. xf(x)F(x) 0 1 2 .25 .50 .25 .25 .75 1.00 0 1 2 m = (0)(.25) + (1)(.50) + (2)(.25) = 1 s2 = (0-1)2(.25) + (1-1)2(.50) + (2-1)2(.25) = .5 xfrfcfrcf 0 1 2 29 45 26 100 .29 .45 .26 29 74 100 .29 .74 1.00

  10. xf(x)F(x) 0 1 2 .25 .50 .25 .25 .75 1.00 0 1 2 m = (0)(.25) + (1)(.50) + (2)(.25) = 1 s2 = (0-1)2(.25) + (1-1)2(.50) + (2-1)2(.25) = .5 xfrfcfrcf 0 1 2 29 45 26 100 .29 .45 .26 29 74 100 .29 .74 1.00

  11. xf(x)F(x) 0 1 2 .25 .50 .25 .25 .75 1.00 0 1 2 m = (0)(.25) + (1)(.50) + (2)(.25) = 1 s2 = (0-1)2(.25) + (1-1)2(.50) + (2-1)2(.25) = .5 xfrfcfrcf 0 1 2 29 45 26 100 .29 .45 .26 29 74 100 .29 .74 1.00

  12. xf(x)F(x) 0 1 2 .25 .50 .25 .25 .75 1.00 0 1 2 m = (0)(.25) + (1)(.50) + (2)(.25) = 1 s2 = (0-1)2(.25) + (1-1)2(.50) + (2-1)2(.25) = .5 xfrfcfrcf 0 1 2 29 45 26 100 .29 .45 .26 29 74 100 .29 .74 1.00

  13. xf(x)F(x) 0 1 2 .25 .50 .25 .25 .75 1.00 0 1 2 m = (0)(.25) + (1)(.50) + (2)(.25) = 1 s2 = (0-1)2(.25) + (1-1)2(.50) + (2-1)2(.25) = .5 xfrfcfrcf 0 1 2 29 45 26 100 .29 .45 .26 29 74 100 .29 .74 1.00

  14. xf(x)F(x) 0 1 2 .25 .50 .25 .25 .75 1.00 0 1 2 m = (0)(.25) + (1)(.50) + (2)(.25) = 1 s2 = (0-1)2(.25) + (1-1)2(.50) + (2-1)2(.25) = .5 xfrfcfrcf 0 1 2 29 45 26 100 .29 .45 .26 29 74 100 .29 .74 1.00 S2 = 0.5546

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