Question

The length of time Jim and Lisa take to complete a 20km marathon has been found to be both normally and independently distributed. If Jim's completion time of the race averages 1.5 hours with a standard deviation of 0.4 hours and Lisa's completion time averages 1.3 hours with a standard deviation of 0.6 hours, determine:

1) The probability that Lisa takes less than 1 hour and 45 minutes to complete a race.

2) The probability that Jim finishes the race at least 10 minutes faster than Lisa.

Answer 1

1)

Let X be the time Lisa takes to complete the race.

Probability that Lisa takes less than 1 hour and 45 minutes to complete a race = P(X < 1 hour 45 minutes)

= P(X < 1.75 hr)

= P[Z < (1.75 - 1.3) / 0.6]

= P[Z < 0.75]

= 0.7734

2)

Let Y be the time Jim takes to complete the race.

Probability that Jim finishes the race at least 10 minutes faster than Lisa. = P(X - Y > 10 minutes)

= P(X - Y > 1/6 hour) = P(X - Y > 0.1667)

If X and Y are normal random variable, X-Y (linear combination of X and Y) is also normal random variable.

E(X - Y) = E(X) - E(Y) = 1.3 - 1.5 = -0.2

Var(X - Y) = Var(X) + Var(Y) = 0.6² + 0.4² = 0.52

Standard deviation of X-Y = = 0.7211

Thus, X-Y ~ N(-0.2, 0.52)

P(X - Y > 0.1667) = P[Z > (0.1667 - (-0.2))/0.7211]

= P[Z > 0.5085]

= 0.3055