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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?
Solution
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):}`
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