Math 462/562 - Part 3 - Assignment 3

Due:Monday, June 20. Nothing accepted after Wednesday, June 22. 10% off for being late. Please work by yourself. See me if you need help.

1.A random variable X has a cumulative distribution function given by

G(a) =

a.(1 points) Find the probability density function for X.

b.(1 points) Find the probability that a given observation of the value of X is greater than or equal to 0.5.

c.(1 points) Find the mean of X.

2.A random variable T has a probability density function given by

f(t) =

a.(1 points) Find k.

b.(1 points) Find the probability that in a given experiment the value of T is greater than or equal to 0.25 and less than or equal to 1.

c.(1 points) Find the cumulative distribution function of T. You should have separate formulas for t < 0.5 and t 0.5.

3.The time between successive cars on a certain road is exponentially distributed and the probability is ½ that the next car will arrive within two minutes. Assume the time between any particular pair of cars is independent of the times between all other pairs of cars.

a.(1 points) What is the probability the next car will arrive within one minute?

b.(1 points) What is the expected time until the next car will arrive?

c.(1.5 points) What is the probability that the 3rd car arrives within 6 minutes?

d.(1.5 points) What is the probability that exactly 3 cars arrive in the next 6 minutes?

4.(4 points) John is going to eat at McDonald's. The time of his arrival is uniformly distributed between 6 and 7 p.m. and it takes him 15 minutes to eat. Mary is also going to eat at McDonald's. The time of her arrival is uniformly distributed between 6:30 and 7:15 and it takes her 25 minutes to eat. Suppose the times of their two arrivals are independent of each other. What is the probability that there will be some time that they are both at McDonald's, i.e. their times at McDonald's overlap. A random variable is uniformly distributed between a and b if the density function is equal to 1/(b-a) between a and b and equal to zero for t less than a or greater than b.