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Chapter 5: Discrete Probability Distributions

Chapter 5: Discrete Probability Distributions. Section 5.3: Mean, Variance, and Standard Deviation of a Probability Distribution. Part 1: Finding the mean. Use the symbol µ as the symbol for mean. Formula for Mean: µ = Σ (x ∙ P(x)). Where x is the outcome and P(x) is the probability

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Chapter 5: Discrete Probability Distributions

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  1. Chapter 5: Discrete Probability Distributions Section 5.3: Mean, Variance, and Standard Deviation of a Probability Distribution

  2. Part 1: Finding the mean Use the symbol µ as the symbol for mean Formula for Mean: µ = Σ (x ∙ P(x)) Where x is the outcome and P(x) is the probability of that outcome (Multiply each outcome by its probability, then add all of the sums together, like in the numerator of weighted mean)

  3. The following is a Probability distribution of the number Of miles ran each day for a particular Individual training for a race

  4. To find mean: µ = ΣX ∙ P(X) µ = ΣX ∙ P(X) = 4 (.2) + 5 (.175) + 6 (.225) + 7 (.4) µ = .8 + .875 + 1.35 + 2.8 µ = 5.83

  5. Part 2: Finding the variance and standard deviation Use the symbol σ2as the symbol for varianceand σ for standard deviation Formula for Variance: σ2= Σ[x2∙ P(x)] - µ2 (Square each outcome then multiply by its probability, then add all of the sums together, then subtract the mean squared)

  6. Part 2: Finding the variance and standard deviation Use the symbol σ2as the symbol for varianceand σ for standard deviation Formula for Standard Deviation: σ= √σ2 (Square root of the variance)

  7. To find variance: σ2 = Σ [X2 ∙ P(X)] - µ2 σ2= 42 (.2) + 52 (.175) + 62 (.225) + 72 (.4) – 5.832 σ2 = 3.2 + 4.375 + 8.1 + 19.6 - 33.9889σ2= 1.2861 ≈ 1.29 To find standard deviation:σ = √σ2σ= √1.29 σ= 1.14

  8. Example 1: Find the variance and standard Deviation forthe number of tails when tossing 3 coins. H T T H T H H T H T H T H T Sample Space: { HHH, THH, HTH, TTH, HHT, THT, HTT, TTT}

  9. Example 1: Find the variance and standard Deviation for the number of tails when tossing 3 coins. Sample Space: { HHH, THH, HTH, TTH, HHT, THT, HTT, TTT} 0 1 2 3 1/8 3/8 3/8 1/8

  10. Example 1: 0 1 2 3 1/8 3/8 3/8 1/8 To find mean: µ = ΣX ∙ P(X) µ = ΣX ∙ P(X) = 0 (1/8) + 1 (3/8) + 2 (3/8) + 3 (1/8) µ = 0 + .375 + .75 + .375 µ = 1.5

  11. Example 1: 0 1 2 3 1/8 3/8 3/8 1/8 To find variance: σ2 = Σ [X2 ∙ P(X)] - µ2 σ2= 02 (1/8) + 12 (3/8) + 22 (3/8) + 32 (1/8) – 1.52 σ2 = 0 + .375 + 1.5 + 1.125 – 2.25σ2= 0.75 To find standard deviation:σ = √σ2σ= √0.75 σ= 0.87

  12. Example 2: Five balls numbered 0, 2, 4, 6 And 8 are placed in a bag. Each time after One is selected and recorded it is replaced. Find the mean, variance and Standard deviation 0 2 4 6 8 1/5 1/5 1/5 1/5 1/5

  13. Example 2: 0 2 4 6 8 1/5 1/5 1/5 1/5 1/5 To find mean: µ = ΣX ∙ P(X) µ = 0 (1/5) + 2 (1/5) + 4 (1/5) + 6 (1/5) + 8 (1/5) µ = 0 + .4 + .8 + 1.2 + 1.6µ = 4

  14. Example 2: 0 2 4 6 8 1/5 1/5 1/5 1/5 1/5 To find variance: σ2 = Σ [X2 ∙ P(X)] - µ2 σ2= 02(1/5) + 22(1/5) + 42(1/5) + 62(1/5) +82(1/5) –42 σ2 = 0 + .8 + 3.2 + 7.2 + 12.8 – 16σ2= 8 To find standard deviation:σ = √σ2σ= √8 σ= 2.83

  15. Try one on your own: The probability distribution of the Number of siblings for high school Students attending a large school is:

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