# Help me understand the (continuous) uniform distribution

I think I didn't pay attention to uniform distributions because they're too easy. So I have this problem

1. It takes a professor a random time between 20 and 27 minutes to walk from his home to school every day. If he has a class at 9.00 a.m. and he leaves home at 8.37 a.m., find the probability that he reaches his class on time.

I am not sure I know how to do it. I think I would use $F(x)$, and I tried to look up how to figure it out but could only find the answer $F(x)=(x-a)/(b-a)$.

So I input the numbers and got $(23-20)/(27-20)$ which is $3/7$ but I am not sure that is the corret answer, though it seems right to me.

I'm not here for homework help (I am not being graded on this problem or anything), but I do want to understand the concepts. Too often I just learn how to do math and don't "really" understand it.

So I would like to know how to properly do uniform distribution problems (of continuous variable) and maybe how to find the $F(x)$. I thought it was the integral but I didn't get the same answer.

Remember I want to understand this. Thanks so much for your time.

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Your arrival time is at a constant interva; $[a,b]$ and the uniform distribution gives, $\int_{a}^{x} \frac{dt}{27-20}$ Your starting point, 8:37, is your time 0 and you want to make it to your class by 9. Your minimum walking time is 20 mins which would give make you still on time. But for a walking time of more than 23 mins you will be late. Hence we want our walking time in $[20,23]$ And these are the bounds of our integral. Hence desired probability is $3/7$
• Your walk time ranges between 20 and 27 minutes in a uniform distribution (flat, not bell curve). Examine the probability by thinking about the area under the curve. Since the only possibilities are from 20 to 27, your graph will be uniform between 20 and 27 and 0 elsewhere. Also, it is important that the total area under the curve is 1 (or 100%). So to find the probability you make it on time, you must find the area between 20 and 23. Since the function is uniform, it follows that the answer is $\frac {23-20} {27-20} = \frac 3 7$.
• You can solve for area using calculus, $\int_{20}^{23} \frac 1 {27-20} dx = \frac 3 7$, but since it is a uniform distribution understanding the area doesn't really require calculus.
• If you don't like area, think about it this way. Ignore the 20 minutes since it doesn't really change the problem. 0-3 minutes means you make it on time, and if you are even a millisecond over 3 minutes you will be late. So 0-3 minutes on time, 3.00... to 7 minutes you will be late. In the realm of possibilities, you have a 3 minute margin for being on time, and a 4 minute margin for being late. 3 acceptable outcomes out of 7 total outcomes means $\frac 3 {3+4} = \frac 3 7$