     1. Chapter 13 Class 12 Probability
2. Serial order wise
3. Ex 13.4

Transcript

Ex 13.4, 13 Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X. Here, X = Sum of numbers on two die Since pair of die are thrown, So, X can be 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Thus, Probability distribution is Now, we need to find variance Var 𝑋﷯ = 𝐸 𝑋﷮2﷯﷯− 𝐸 𝑋﷯﷯﷮2﷯ Finding E(X) E(X) = 𝑖 = 1﷮𝑛﷮𝑥𝑖𝑝𝑖﷯ = 2 × 1﷮36﷯+3 × 2﷮36﷯+4 × 3﷮36﷯+5 × 4﷮36﷯+6 × 5﷮36﷯+7 × 6﷮36﷯ +8 × 5﷮36﷯+9 × 4﷮36﷯+10 × 3﷮36﷯+11 × 2﷮36﷯+12 × 1﷮36﷯ = 2 + 6 + 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12﷮36﷯ = 252 ﷮36﷯ = 7 Thus E(x) = 7 Finding E 𝑿﷮𝟐﷯﷯ E 𝑿﷮𝟐﷯﷯= 𝑖=1﷮𝑛﷮ 𝑥﷮2﷯𝑖𝑃𝑖﷯ = 2﷮2﷯× 1﷮36﷯ + 3﷮2﷯× 2﷮36﷯ + 4﷮2﷯ × 3﷮36﷯+ 5﷮2﷯× 4﷮36﷯+ 6﷮2﷯× 5﷮36﷯+ 7﷮2﷯× 6﷮36﷯ + 8﷮2﷯ × 5﷮36﷯+ 9﷮2﷯× 4﷮36﷯+ 10﷮2﷯× 3﷮36﷯+ 11﷮2﷯ × 2﷮36﷯+ 12﷮2﷯× 1﷮36﷯ = 4 + 18 + 48 + 100 + 180 + 294 + 320 + 324 + 300 + 242 + 144﷮36﷯ = 1974﷮36﷯ = 𝟑𝟐𝟗﷮𝟔﷯ Now, Var 𝑿﷯ = 𝑬 𝑿﷮𝟐﷯﷯− 𝑬 𝑿﷯﷯﷮𝟐﷯ = 329﷮6﷯− 7﷯﷮2﷯ = 329 − 6 × 49﷮6﷯ = 35﷮6﷯ = 5.833 Standard deviation is given by 𝝈﷮𝒙﷯= ﷮𝒗𝒂𝒓 𝑿﷯﷯ = ﷮ 35﷮6﷯﷯ = ﷮5.833﷯ = 2.415

Ex 13.4 