Question

Problem 1 A subway train on the #5 line arrives every eight minutes. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution In each appropriate box you are to enter either a rational number in "p/" format or a decimal value accurate to the nearest 0.01 a. The waiting time is modeled by a random variable X with X (pick on distribution b. The density function for X is given by f(x) = with SXS c. The mean Hx = d. The standard deviation Ox = e. The probability that the commuter waits less than one minute is

Answer 1

Solution:

Problem 1.

a. As already indicated in the question, the waiting time follows uniform distribution. Also, as the subway train comes in every 8 mins, a person has to wait maximum for 8 minutes (minimum is of course 0 minutes).

So, X ~ uniform (0, 8) distribution.

b. Density function is: (1/(B - A)) = (1/(8 - 0)) = 1/8 for0 <= X <= 8; 0 otherwise [A signifies minimum of interval and B signifies maximum of interval].

c. For a uniform distribution, mean = (A + B)/2

Mean = (0 + 8)/2 = 4 mins

d. For a uniform distribution, variance = (B - A)²/12

So, standard deviation = (B - A)/12^0.5 = (8-0)/12^0.5 = 2.31 approx.

e. Prob(X < 1) = cumulative frequency calculated as: F(X<=1) = (X - A)/(B - A)

Prob(X < 1) = (1 -0)/(8-0) = 1/8 or 0.125