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Example 28 - Find variance of number obtained on throw of die - Variance and Standard Deviation of a Random Variable

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  1. Class 12
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Example 28 Find the variance of the number obtained on a throw of an unbiased die. Let X be number obtained on a throw So, value of X can be 1, 2, 3, 4, 5 or 6 Since die unbiased, Probability of getting of each number is equal P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = 1﷮6﷯ Hence, probability distribution The mean Expectation value is given by E(X) = 𝒊 = 𝟏﷮𝒏﷮𝒙𝒊𝒑𝒊﷯ = 1 × 1﷮6﷯+2 × 1﷮6﷯+ 3 × 1﷮6﷯+ 3 × 1﷮6﷯+ 5 × 1﷮6﷯+ 6 × 1﷮6﷯ = 21﷮6﷯ The variance of x is given by : Var 𝑿﷯=𝑬 𝑿﷮𝟐﷯﷯− 𝑬 𝑿﷯﷯﷮𝟐﷯ So, finding 𝐸 𝑋﷮2﷯﷯ E 𝑋﷮2﷯﷯= 𝑖 = 1﷮𝑛﷮ 𝑥﷮𝑖﷯﷮2﷯𝑝𝑖﷯ = 12 × 1﷮6﷯+22 × 1﷮6﷯+ 32 × 1﷮6﷯+ 42 × 1﷮6﷯+ 52 × 1﷮6﷯+ 62 × 1﷮6﷯ = 1 + 4 + 9 + 16 + 25 + 36﷮6﷯ = 91﷮6﷯ Now, Var 𝑋﷯=𝐸 𝑋﷮2﷯﷯− 𝐸 𝑋﷯﷯﷮2﷯ = 91﷮6﷯− 21﷮6﷯﷯﷮2﷯ = 91﷮6﷯− 441﷮36﷯ = 546 − 441﷮36﷯ = 105﷮36﷯ = 35﷮12﷯ Hence, variance is 𝟑𝟓﷮𝟏𝟐﷯

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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