Question

Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function

F(x) = `{{:(0",",  "for"  x ≤ 0),(x/2",",  "for"  0 ≤ x < 1),(1/2",",  "for" ≤ x < 2),(x/4",",  "for"  2 ≤ x < 4),(1",",  "for"  x ≥ 4):}` 
Is the distribution function continuous? If so, give its probability density function?

Answer 1

Yes, the distribution function is continuous on [0, 4]

The probability density function
f(x) = `("d"["f"(x)])/("d"x) = {{:(0",",  "for"  x < 0),(1/2",",  "for"  0 ≤ x ≤ 1),(1/4",",  "for"  2 ≤ x < 4),(0",",  "for"  x ≥ 4):}`