Ex 13.4, 12

Transcript

Ex 13.4, 12 Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X). The first 6 positive integers are : 1, 2, 3, 4, 5, 6 Let X : be the larger number of two numbers selected The possible outcomes are Sample space = S = 1, 2﷯, 1, 3﷯, 1, 4﷯, 1, 5﷯, 1, 6﷯,﷮ 2, 1﷯, 2, 3﷯, 2, 4﷯, 2, 5﷯, 2, 6﷯,﷯﷮ 3, 1﷯, 3, 2﷯, 3, 4﷯, 3, 5﷯, 3, 6﷯,﷮ 4, 1﷯, 4, 2﷯, 4, 3﷯, 4, 5﷯, 4, 6﷯,﷯﷮ 5, 1﷯, 5, 2﷯, 5, 3﷯, 5, 4﷯, 5, 6﷯,﷮ 6, 1﷯, 6, 2﷯, 6, 3﷯, 6, 4﷯, 6, 5﷯﷯﷯﷯ Total number of possible outcomes = 30 The larger number can be : 2, 3, 4, 5 or 6 So, the values of X can be : 2, 3, 4, 5 or 6 Thus, the probability distribution is The mean Expected value is given by 𝝁=𝑬 𝑿﷯= 𝑖=1﷮𝑛﷮𝑥𝑖𝑃𝑖﷯ = 2 × 2﷮30﷯+3 × 4﷮30﷯+ 4 × 6﷮30﷯+ 5 × 8﷮30﷯+ 6 × 10﷮30﷯ = 4 + 12 + 24 + 40 + 60﷮30﷯ = 140﷮30﷯ = 𝟏𝟒﷮𝟑﷯