A-Level Discrete random variables: worked solution

Question

With C ~ N(8, 0.2^2) and T ~ N(17, 1.4^2) independent, find the probability her total cycling time over five random days exceeds twice her train time on one random day.

Method: a sum or difference of independent normal variables is itself normal. Build ONE new variable so that the comparison turns into a simple question about it being positive. Step 1 - define the variable. The five days are five separate, independent observations of C, so write them as C_1, C_2, C_3, C_4, C_5. The train time on one day is T. Let X = C_1 + C_2 + C_3 + C_4 + C_5 - 2T The event we want, total cycling time exceeds twice the train time, is exactly X > 0. Step 2 - find the mean. Expectation is linear, so E(X) = 5E(C) - 2E(T) = 5(8) - 2(17) = 40 - 34 = 6 Step 3 - find the variance. For independent variables the variances ADD, and a constant multiplier comes out SQUARED. Note the difference between five separate observations and five times one observation: five separate C's contribute 5 x Var(C), not 5² x Var(C); the -2T contributes (-2)² x Var(T) = 4 x Var(T), and the minus sign disappears on squaring. Var(X) = 5(0.2²) + 4(1.4²) = 5(0.04) + 4(1.96) = 0.2 + 7.84 = 8.04 So X ~ N(6, 8.04). Step 4 - standardise and read off the probability. P(X > 0) = P(Z > (0 - 6)/sqrt(8.04)) = P(Z > -6/2.8355) = P(Z > -2.116) = 0.9828 P(total cycling time exceeds twice the train time) = 0.983 (3 s.f.)