Probability and Random Variables – Cumulative Distribution Functions

Video transcript

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2. Now the CDF tells you the probability for
3. random variable x to be below a certain value.
4. So in our case over here, it’s 14.
5. Okay, so this area under here, that’s the probability that
6. the random variable’s less than or equal to 14.
7. And the CDF function is written just like this.
8. F(14), that’s the probability of the random variable to take
9. on a value below 14.
10. And I can write it more generally, so
11. if I can write it for a general x.
12. So for any of amount of coffee x that Steve might drink,
13. the CDF gives the probability to drink less than that value.
14. Now, does it matter whether I write less than or
15. equal to here or just less than?
16. And the answer is no, it doesn’t matter,
17. because the probability that he’ll drink
18. exactly that value x down to the molecule is 0.
19. For continuous distributions, doesn’t matter whether this is
20. strict equality or less than or equal to.
21. So this subtraction that we were doing earlier,
22. this can actually be written in terms of the CDFs.
23. So if we want the probability that X is between 10 and 14,
24. it’s the CDF at 14 minus the CDF at 10.
25. And I can write it like this.
26. And if I want the probability that Steve is gonna drink more
27. than a certain value, like 11 liters, then I can actually
28. get that by remembering that the area under the whole thing
29. is 1 and subtracting the area to the left of 11 here.
30. Okay, so what is the smallest possible value for the CDF?
31. And at 0 all the way on the left, okay?
32. So if X is -10, that’s way over here, then F is about 0.
33. Now what about 0.25, okay?
34. So if X is 8, then the area under here is about 0.25.
35. This is about a quarter of the area under the whole thing.
36. That’s a quarter probability to get less than 8.
37. Then what about 10?
38. F(10) is about 0.5,
39. because that’s really where the center of the distribution is.
40. F(12) is about 0.75 and then F(20) is about 1,
41. so if I actually plot out what F looks like, it looks like this.
42. So it’s 0 all the way to the left, and
43. then when I get to about 8, it’s 0.25.
44. When I get to 10 it’s about 0.5, I get up to 12 it’s about 0.75,
45. and when I get up to 20, it’s 1, okay.
46. So now you
47. see I seem to be drawing bumps very often as my example of what
48. a PDF ought to look like, which means the CDF
49. always ends up looking kinda like that function down here.
50. Now, there is a very good reason that I keep drawing bell curves
51. like this and I will tell you what it is.
52. It is called the central limit theorem and that is up next.
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