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DISCRETE RANDOM VARIABLES Monday 25 th February 2013 Learning objectives: To understand probability distributions for discrete random variables. To be able to find the mean (expected value), variance and standard deviation of discrete random variables.

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Discrete random variable A discrete random variable is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4,... Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten.

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Probability distributions X represents the score when a dice is rolled. X is a discrete random variable. Score on dice, x 123456 P(X=x) Σ P(X=x) = 1 If X is a discrete random variable with probability function P(X=x) or p(x) then:

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Cumulative distribution functions Score on dice, x 123456 P(X

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Mean of X or expected value of X Score on dice, x 123456 P(X=x) μ = E(X) = Σ xP(X=x) If X is a discrete random variable with probability function P(X=x) or p(x) then the expected value of X:

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Variance and standard deviation σ 2 = Var(X) = Σ (x – μ) 2 P(X=x) = Σ x 2 P(X=x) – μ 2 The variance of a distribution is the mean of the sum of the squared deviations from the mean. The standard deviation, σ, of a distribution is the positive square root of the variance. σ 2 = Var(x) = E(X 2 ) – μ 2

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Variance and standard deviation Score on dice, X 123456 X2X2X2X2 P(X=x) σ 2 = Var(x) = E(X 2 ) – μ 2

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Listing possibilities: have a system f c fffc cf cc fffffc fcf fcc cff cfc ccf ccc fffffffc ffcf ffcc fcff fcfc fccf fccc cfff cffc cfcf cfcc ccff ccfc cccf cccc 1 stick 2 sticks 3 sticks

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Working out probabilities: f c fffc cf cc fffffc fcf fcc cff cfc ccf ccc fffffffc ffcf ffcc fcff fcfc fccf fccc cfff cffc cfcf cfcc ccff ccfc cccf cccc 1 stick 2 sticks 3 sticks Each stick falls independently, f=0.7c=0.3 1 x P(f,f,f,f) = 0.7 x 0.7 x 0.7 x 0.7= 0.2401 = 0.7 4 4 x P(f,f,f,c)= 0.7 3 x 0.3 x 4= 6 x P(f,f,c,c)=0.7 2 x 0.3 2 x 6 = 4 x P(f,c,c,c) = 0.7 x 0.3 3 x 4 = 1 x P(c,c,c,c) = 0.3 4 =

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Relative frequency The score is an example of discrete random variable. –Let S stand for score. Capital letters are used for random variables. –P(S=3) means ‘the probability that S=3 –P(S=3) = 0.4116 Number of flat sides up. 01234 Score51234 s.12345P(S=s) Probability function 0.07560.26460.41160.24010.0081 Note s is used for individual values of the random variable S

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P(X=x) as a stick/bar graph

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TASK Exercise A – Page 53 & 54 Questions: 1, 2, 3, 5 & 6 Do rest at home.

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Mean, variance and standard deviation If I were to throw 10000 times, I could work out the mean like the below. Multiply each of my probabilities by 10000 and then divide by 10000 s.12345 P(S=s) Probability function 0.07560.26460.41160.24010.0081

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Mean, variance and standard deviation However, multiplying and dividing by 10000 both top and bottom seems unnecessary and it is s.12345 P(S=s) Probability function 0.07560.26460.41160.24010.0081

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MEAN of: Discrete random Variables Mean S =Σs x P(S=s) The mean of a random variable is usually denoted by μ (‘mu’) Task B1, B2

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VARIANCE Mean = 0x0.15 + 1x0.25 + 2x0.25 + 3x0.25 + 4x0.1=1.9 x01234 P(X=x) Probability function 0.150.250.250.250.1 x x-μ (x-μ) 2 P(X=x) (x-μ) 2 x P(X=x) 0-1.93.610.150.5415 1-0.90.810.250.2025 20.10.010.250.0025 31.11.210.250.3025 42.14.410.10.4410

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Variance & Standard deviation x x-μ (x-μ) 2 P(X=x) (x-μ) 2 x P(X=x) 0-1.93.610.150.5415 1-0.90.810.250.2025 20.10.010.250.0025 31.11.210.250.3025 42.14.410.10.4410 Variance = σ 2 1.49 Standard deviation = σ 1.22 The standard deviation or random variables is normally denoted as σ

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TASK Page 56 question 2 Homework – test yourself

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Mean and Standard Deviation of Discrete Random Variables.

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