continuous uniform distribution question

CFA Level I
#1

example - consider a random variable X that follows a continuous uniform distribution 8 < X < 20 …

I don’t understand why F(21) = 1.

In my mind i read it as X cannot be greater than or equal to 20. So why would it be 1 (100% ??) if F(21)??

#2

F(21) = P(X ≤ 21)

As X cannot be greater than 20, it is _ always less than or equal to_ 21, so F(21) = 1.

Some more values of F:

  • F(−1,000) = P(X ≤ −1,000) = 0
  • F(−10) = P(X ≤ −10) = 0
  • F(0) = P(X ≤ 0) = 0
  • F(1) = P(X ≤ 1) = 0
  • F(4) = P(X ≤ 4) = 0
  • F(8) = P(X ≤ 8) = 0
  • F(9) = P(X ≤ 9) = 1/12
  • F(10) = P(X ≤ 10) = 2/12 = 1/6
  • F(11) = P(X ≤ 11) = 3/12 = 1/4
  • F(14) = P(X ≤ 14) = 6/12 = 1/2
  • F(19) = P(X ≤ 19) = 11/12
  • F(20) = P(X ≤ 20) = 12/12 = 1
  • F(21) = P(X ≤ 21) = 1
  • F(100) = P(X ≤ 100) = 1
  • F(1,000,000) = P(X ≤ 1,000,000) = 1

The graph of F(X) looks like this:

__________/¯¯¯¯¯¯¯¯¯¯

The left part of the graph is at zero (vertically); the right part is at 1. The kink on the bottom is at X = 8; the kink on the top is at X = 20.

#3

“As X cannot be greater than 20, it is _ always less than or equal to_ 21” Ok great so that’s my thinking and I understand that part… But what I dont understand is what is the true meaning in words behind F(21) = 1?

I’m thinking in probability terms and if something is equal to 1 there is a 100% probability. Clearly that’s wrong

#4

That’s exactly what it means, and clearly it’s right.

There’s a 100% probability that X is less than or equal to 21; it’s _ always _ less than or equal to 21.

#5

Ok I think I understand now. But what would be the difference then between

  • F(1) = P(X ≤ 1) = 0 and F(21) = 1.

I guess I’m really getting stuck on what it means by the difference of =0 vs = 1 ?

#6

Because in my mind from the first probability function of 8 < X < 20 … neither of the above can happen. but what is the difference in notation of 0 vs. 1

#7

Let’s take a look at a few possible values for X: 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20.

Consider F(1) = P(X ≤ 1), and test it for all of these possible values of X:

  • X = 8. Is X ≤ 1? Nope.
  • X = 9. Is X ≤ 1? Nope.
  • X = 10. Is X ≤ 1? Nope.
  • X = 11. Is X ≤ 1? Nope.
  • X = 12. Is X ≤ 1? Nope.
  • X = 13. Is X ≤ 1? Nope.
  • X = 14. Is X ≤ 1? Nope.
  • X = 15. Is X ≤ 1? Nope.
  • X = 16. Is X ≤ 1? Nope.
  • X = 17. Is X ≤ 1? Nope.
  • X = 18. Is X ≤ 1? Nope.
  • X = 19. Is X ≤ 1? Nope.
  • X = 20. Is X ≤ 1? Nope.

In fact, X is _ never _ less than or equal to 1, so P(X ≤ 1) = 0, so F(1) = 0.

Consider F(21) = P(X ≤ 21), and test it for all of these possible values of X:

  • X = 8. Is X ≤ 21? Yup.
  • X = 9. Is X ≤ 21? Yup.
  • X = 10. Is X ≤ 21? Yup.
  • X = 11. Is X ≤ 21? Yup.
  • X = 12. Is X ≤ 21? Yup.
  • X = 13. Is X ≤ 21? Yup.
  • X = 14. Is X ≤ 21? Yup.
  • X = 15. Is X ≤ 21? Yup.
  • X = 16. Is X ≤ 21? Yup.
  • X = 17. Is X ≤ 21? Yup.
  • X = 18. Is X ≤ 21? Yup.
  • X = 19. Is X ≤ 21? Yup.
  • X = 20. Is X ≤ 21? Yup.

In fact, X is _ always _ less than or equal to 21, so P(X ≤ 21) = 1, so F(21) = 1.

#8

GOT IT! Thank you very much!

#9

My pleasure.