Question

The Women's Tennis Association finds that the amount of time between two points in a tennis match is uniformly distributed between 1.5 and 4 minutes. 1. Find the probability that the time between two points lasts more than 2 minutes 2. Find the probability that the time between two points lasts less than 3 minutes 3. Find the 30th percentile of times between points. 4. Give a lower bound on the time between two points for the top 25% longest times between points

Answer 1

Formulas: 1) Uniform distribution: If X is a uniform random variable then probability density function of X is f(x) = for a sxsb. = 0 otherwise Let X = The amount of time between two points in minutes Given that X follows unifrom distribution between 1.5 minutes and 4 minutes. = f(x) = -1.5 = 25 for 1.5 5x54

1) Now we have to find the probability that the time between two points lasts more than 2 minutes. i.e we have to find P(x > 2). P(X > 2) = 5* f(x) dx = Sam zis dx = 25 (S* dx) = zs (7) = ks (4 – 2) = 0.8 : P(X > 2) = 0.8

2) Now we have to find the probability that the time between two points lasts less than 3 minutes. i.e we have to find P(X<3). P(X < 3) = 51.5 f(x) dx = 615 2. dx = . (1.5 dx) = } (x)is 1(3 – 1.5) :: P(X < 3) = 0.6

Now we have to find the 30th percentile of times between two points. i. e we have to find the value of x such that P(X < x) = 30% = 0.3. P(X < x) = 55(x) dx = 15 2. dx = 2. (S15 dx) = (x).5 = 2.(x – 1.5) P(X < x) = 0.3 = *7.5 = 0.3= x = 0.3 + 2.5 +1.5 = 2.25 C : x = 2.25

4) Now we have to find the lower bound on the time between two points for the top 25% longest times between two points. i. e we have to find the value of x such that P(X> x) = 25% = 0.25. P(X > x) = * f(x) dx = * idx = 2 (&* dx) = 25 () = zs (4 – x) P(X < x) = 0.25=4*= 0.25 = x= 4 – 2.5 * 0.25 = 3.375 :: x = 3.375