1. Continuous Random Variables
Lecture 36
Section 7.5.3
Mon, Mar 29, 2004

2. Continuous Probability Distribution Functions
Continuous Probability Distribution Function (pdf) – A function such that the area under the curve over an interval a ≤ x ≤ b equal the probability that a ≤ X ≤ b.
In other words, area = probability.

3. Example
The TI-83 will return a random number between 0 and 1 if we enter rand and press ENTER.
Let X be a random number from 0 to 1.
Then X has a uniform distribution from 0 to 1.

4. Example
The graph of the pdf of X.
f(x) 1 x 1

5. Example
What is the probability that the random number is at least 0.3?

6. Example
What is the probability that the random number is at least 0.3?
f(x) 1 x
0.3 1

7. Example
What is the probability that the random number is at least 0.3?
f(x) 1
Area = 0.7 x
0.3 1

8. Example
What is the probability that the random number is between 0.3 and 0.8?

9. Example
What is the probability that the random number is between 0.3 and 0.8?
f(x) 1 x
0.3
0.8 1

10. Example
What is the probability that the random number is between 0.3 and 0.8?
f(x) 1
Area = 0.5 x
0.3
0.8 1

11. Example
Now suppose we use the TI-83 to get two random numbers from 0 to 1, and then add them together.
Let Y = the sum of the two random numbers.
What is the pdf of Y?

12. Example
The graph of the pdf of Y.
f(y) 1 y 1 2

13. Example
The graph of the pdf of Y.
f(y) 1
Area = 1 y 1 2

14. Example What is the probability that Y is between 0.5 and 1.5? f(y) 1
0.5 1
1.5 2

15. Example
The probability equals the area under the graph from 0.5 to 1.5.
f(y) 1 y
0.5 1
1.5 2

16. Example Cut it into two simple shapes. The total area is 0.75. f(y) 1
0.5 1
1.5 2

17. Example
Suppose we get 12 random numbers between 0 and 1 from the TI-83 and add them all up.
Let X = sum of 12 random numbers from 0 to 1.
What is the pdf of X?

18. Example
It turns out that the pdf of X is approximately normal with a mean of 6 and a standard of 1. 6 7 8 9 5 4 3
N(6, 1) x

19. Example What is the probability that the sum will be between 5 and 7?
P(5 < X < 7) = P(-1 < Z < 1)
= – =

20. Example What is the probability that the sum will be between 4 and 8?
P(4 < X < 8) = P(-2 < Z < 2)
= – =

21. Experiment
Use the TI-83 to see if we really do get a total between 5 and 7 about 68% of the time.
Enter the expression
sum(rand(12))
and press ENTER.
Do it several times and see how often the total is between 5 and 7.

22. Experiment
We should see a value between 5 and 7 about 68% of the time.
We should see a value between 4 and 8 about 95% of the time.
We should see a value between 3 and 9 nearly always (99.7%).

23. Exercise Roll a die 10 times and let X = the sum of the 10 numbers.
Fact: X is approximately normal with mean 35 and standard deviation 5.4.

24. Exercise What is the probability that the sum will be at least 45?
Between 20 and 50? 35
N(35, 5.4) x

25. Assignment
Page 442: Exercises 64 – 69.
Page 451: Exercises 104 – 106.