Suppose three particles are particle projected, P, Q and R. Each particle is projected once. The projection is modelled as an exponentially distributed random variable. Assuming that the average projections for P, Q and R are 50m, 60m and 70m respectively, find:
(a) Probability of a particle being projected 150m or further.
(b) Probability that P will go further than R.
If we call
then the exponential probability distribution
(which is normalized so the total probability is 1, i.e., the integral from 0 to infinity of the probability distribution is 1.)
The probability of an x >= 150 should be:
The probability that one particle goes farther than another should be
in other words choose some value for the variable with the P distribution (i.e., y). Find the probability for that y. Now multiply that by the probability that the other variable has a value less than that. Now integrate over all possible values of y. Note is the variable related to , and is the variable related to
As for a hint on part c), you don't need to worry about the winner, just that one particle beats the world record. When you are working with the maximum, it is often better to calculate .