1. Continuous Random Variables
Lecture 25
Section 7.5.3
Fri, Oct 29, 2004

2. Continuous Probability Distribution Functions
Continuous Probability Distribution Function (pdf) – For a random variable X, it is a function with the property that the area between the graph of the function and an interval a ≤ x ≤ b equals 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.
These numbers have a uniform distribution from 0 to 1.
Let X be the random number returned by the TI-83.

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 x
0.3 1

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

9. Example
What is the probability that the random number is at least 0.3?
Probability = 70%.
f(x) 1
Area = 0.7 x
0.3 1

10. 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

11. 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

12. 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

13. Example
What is the probability that the random number is between 0.3 and 0.8?
Probability = 50%.
f(x) 1
Area = 0.5 x
0.3
0.8 1

14. Experiment Use the TI-83 to generate 500 values of X.
Use rand(500) to do this.
Check to see what proportion of them are between 0.3 and 0.8.
Use a TI-83 histogram and Trace to do this.

15. 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?

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

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

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

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

20. 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

21. Example Cut it into two simple shapes, with areas 0.25 and 0.5. f(y) 1
0.5 1
1.5 2

22. Example The total area is 0.75. The probability is 75%. Area = 0.75
f(y) 1
Area = 0.75 y
0.5 1
1.5 2

23. Verification Use the TI-83 to generate 500 values of Y.
Use rand(500) + rand(500).
Use a histogram to find out how many are between 0.5 and 1.5.

24. Example
Suppose we get 12 random numbers, uniformly distributed 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?

25. 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

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

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

28. 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 500 times and see how often the total is between 5 and 7.

29. 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%).