Uniform Distribution

Question

The lengths of a professor's classes has a continuous uniform distribution between 50.0 min and 52.0 min. If one such class is randomly selected, find the probability that the class length is between 50.1 and 50.3 min.

P(50.1 < X < 50.3)

Answer 1

Question: The lengths of a professor's classes has a continuous uniform distribution between 50.0 min and 52.0 min. If one such class is randomly selected, find the probability that the class length is between 50.1 and 50.3 min. (i.e., P(50.1 < X < 50.3)).

Answer: This is a uniformly distributed random variable. Let's come up with its PDF: for the Unif(a, b) distribution, the PDF is f(x) = 1/(b-a). Here, a = 50 and b = 52, and I'm using those numbers since the length of the class is itself the uniformly distributed phenomenon we're representing as a random variable X, and those numbers a and b were its associated parameters. So, f(x) = 1/(52-50) = 1/2.

Now, we evaluate the CDF for this random variable X over the times we're interested in, 50.1 and 50.3.

Specifically, since we have an "in-between" problem on our hands, we are evaluating a difference of CDF values, F(50.3) - F(50.1):

F(50.3) - F(50.1) = ∫(1/2)dx, from x = 50.1 to x = 50.3

F(50.3) - F(50.1) = (x/2), from x = 50.1 to x = 50.3.

F(50.3) - F(50.1) = (50.3 - 50.1)/2

F(50.3) - F(50.1) = 0.2/2

F(50.3) - F(50.1) = 0.1

P(50.1 < X < 50.3) = 0.1

I hope this helps!