1. Continuous Random Variables
Lecture 25
Section 7.5.4
Wed, Oct 10, 2007

2. Uniform Distributions
The uniform distribution from a to b is denoted U(a, b).
1/(b – a) a b

3. Hypothesis Testing (n = 1)
An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).
H0: X is U(0, 1).
H1: X is U(0.5, 1.5).
One value of X is sampled (n = 1).

4. Hypothesis Testing (n = 1)
An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).
H0: X is U(0, 1).
H1: X is U(0.5, 1.5).
One value of X is sampled (n = 1).
If X is more than 0.75, then H0 will be rejected.

5. Hypothesis Testing (n = 1)
Distribution of X under H0:
Distribution of X under H1:
0.5 1
1.5
0.5 1
1.5

6. Hypothesis Testing (n = 1)
What are  and ? 1
0.5 1
1.5 1
0.5 1
1.5

7. Hypothesis Testing (n = 1)
What are  and ? 1
0.5
0.75 1
1.5 1
0.5
0.75 1
1.5

8. Hypothesis Testing (n = 1)
What are  and ? 1
0.5
0.75 1
1.5
Acceptance Region
Rejection Region 1
0.5
0.75 1
1.5

9. Hypothesis Testing (n = 1)
What are  and ? 1
0.5
0.75 1
1.5 1
0.5
0.75 1
1.5

10. Hypothesis Testing (n = 1)
What are  and ?
 = ¼ = 0.25 1
0.5
0.75 1
1.5 1
0.5
0.75 1
1.5

11. Hypothesis Testing (n = 1)
What are  and ?
 = ¼ = 0.25 1
0.5
0.75 1
1.5
 = ¼ = 0.25 1
0.5
0.75 1
1.5

12. Example
Now suppose we use the TI-83 to get two random numbers from 0 to 1, and then add them together.
Let X2 = the average of the two random numbers.
What is the pdf of X2?

13. Example
The graph of the pdf of X2.
f(y) ? y
0.5 1

14. Example
The graph of the pdf of X2.
f(y) 2
Area = 1 y
0.5 1

15. Example What is the probability that X2 is between 0.25 and 0.75? f(y)
0.25
0.5
0.75 1

16. Example What is the probability that X2 is between 0.25 and 0.75? f(y)
0.25
0.5
0.75 1

17. Example
The probability equals the area under the graph from 0.25 to 0.75.
f(y) 2 y
0.25
0.5
0.75 1

18. Example Cut it into two simple shapes, with areas 0.25 and 0.5. f(y) 2
0.25
0.5
0.75 1

19. Example
The total area is 0.75.
f(y) 2
Area = 0.75 y
0.25
0.5
0.75 1

20. Verification Use Avg2.xls to generate 10000 pairs of values of X.
See whether about 75% of them have an average between 0.25 and 0.75.

21. Hypothesis Testing (n = 2)
An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).
H0: X is U(0, 1).
H1: X is U(0.5, 1.5).
Two values of X are sampled (n = 2).
Let X2 be the average.
If X2 is more than 0.75, then H0 will be rejected.

22. Hypothesis Testing (n = 2)
Distribution of X2 under H0:
Distribution of X2 under H1:
0.5 1
1.5 2
0.5 1
1.5 2

23. Hypothesis Testing (n = 2)
What are  and ?
0.5 1
1.5 2
0.5 1
1.5 2

24. Hypothesis Testing (n = 2)
What are  and ?
0.5 1
1.5 2
0.75
0.5 1
1.5 2
0.75

25. Hypothesis Testing (n = 2)
What are  and ?
0.5 1
1.5 2
0.75
0.5 1
1.5 2
0.75

26. Hypothesis Testing (n = 2)
What are  and ?
0.5 1
1.5 2
 = 1/8 = 0.125
0.75
0.5 1
1.5 2
0.75

27. Hypothesis Testing (n = 2)
What are  and ?
0.5 1
1.5 2
 = 1/8 = 0.125
0.75
0.5 1
1.5 2
 = 1/8 = 0.125
0.75

28. Conclusion
By increasing the sample size, we can lower both  and  simultaneously.

29. Example
Now suppose we use the TI-83 to get three random numbers from 0 to 1, and then average them.
Let X3 = the average of the three random numbers.
What is the pdf of X3?

30. Example
The graph of the pdf of X3. 3 y
0.25
0.5
0.75 1

31. Example
The graph of the pdf of X3. 3
Area = 1 y
0.25
0.5
0.75 1

32. Example What is the probability that X3 is between 0.25 and 0.75? 3 y
0.25
0.5
0.75 1

33. Example What is the probability that X3 is between 0.25 and 0.75? 3 y
0.25
0.5
0.75 1

34. Example
The probability equals the area under the graph from 0.25 and 0.75. 3
Area = y
0.25
0.5
0.75 1

35. Hypothesis Testing (n = 3)
An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).
H0: X is U(0, 1).
H1: X is U(0.5, 1.5).
Three values of X3 are sampled (n = 3). Let X3 be the average.
If X3 is more than 0.75, then H0 will be rejected.

36. Hypothesis Testing (n = 3)
Distribution of X3 under H0:
Distribution of X3 under H1:
0.5
1.5 1
1.5
0.5 1

37. Hypothesis Testing (n = 3)
Distribution of X3 under H0:
Distribution of X3 under H1:
 =
0.5
1.5 1
 =
1.5
0.5 1

38. Example
Suppose we get 12 random numbers, uniformly distributed between 0 and 1, from the TI-83 and get their average.
Let X12 = average of 12 random numbers from 0 to 1.
What is the pdf of X12?

39. Example
It turns out that the pdf of X12 is nearly exactly normal with a mean of 1/2 and a standard deviation of 1/12.
N(1/2, 1/12) x
1/3
1/2
2/3

40. Example
What is the probability that the average will be between 0.45 and 0.55?
Compute normalcdf(0.45, 0.55, 1/2, 1/12).
We get

41. Experiment
Use the TI-83 to generate 100 averages of 12 random numbers each.
Use
rand(100)  L1
L1 + L2  L2
L2/12  L2
Test the Rule.
12 times