continuous random variables lecture 26 section 7.5.4 mon, mar 5, 2007
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Continuous Random Variables
Lecture 26Section 7.5.4Mon, Mar 5, 2007
Uniform Distributions
The uniform distribution from a to b is denoted U(a, b).
a b
1/(b – a)
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).
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.
Hypothesis Testing (n = 1)
Distribution of X under H0:
Distribution of X under H1:0 0.5 1 1.5
1
0 0.5 1 1.5
1
Hypothesis Testing (n = 1)
What are and ?
0 0.5 1 1.5
1
0 0.5 1 1.5
1
Hypothesis Testing (n = 1)
What are and ?
0.75
0.75
0 0.5 1 1.5
1
0 0.5 1 1.5
1
Hypothesis Testing (n = 1)
What are and ?
0.75
0.75
0 0.5 1 1.5
1
0 0.5 1 1.5
1Acceptance Region Rejection Region
Hypothesis Testing (n = 1)
What are and ?
0.75
0.75
0 0.5 1 1.5
1
0 0.5 1 1.5
1
Hypothesis Testing (n = 1)
What are and ?
0.75
0.75
0 0.5 1 1.5
1
0 0.5 1 1.5
1
= ¼ = 0.25
Hypothesis Testing (n = 1)
What are and ?
0.75
0 0.5 1 1.5
1
0 0.5 1 1.5
1
= ¼ = 0.25
= ¼ = 0.25
0.75
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?
Example
The graph of the pdf of X2.
y
f(y)
0 0.5 1
?
Example
The graph of the pdf of X2.
y
f(y)
0 0.5 1
2
Area = 1
Example
What is the probability that X2 is between 0.25 and 0.75?
y
f(y)
0 0.5 10.25 0.75
2
Example
What is the probability that X2 is between 0.25 and 0.75?
y
f(y)
0 0.5 10.25 0.75
2
Example
The probability equals the area under the graph from 0.25 to 0.75.
y
f(y)
0 0.5
2
10.25 0.75
Example
Cut it into two simple shapes, with areas 0.25 and 0.5.
y
f(y)
0 0.5 10.25 0.75
2
Area = 0.5
Area = 0.250.5
Example
The total area is 0.75.
y
f(y)
0 0.5 10.25 0.75
2
Area = 0.75
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.
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.
Hypothesis Testing (n = 2)
Distribution of X2 under H0:
Distribution of X2 under H1:0 0.5 1 1.5
2
0 0.5 1 1.5
2
Hypothesis Testing (n = 2)
What are and ?
0 0.5 1 1.5
2
0 0.5 1 1.5
2
Hypothesis Testing (n = 2)
What are and ?
0 0.5 1 1.5
2
0 0.5 1 1.5
2
0.75
0.75
Hypothesis Testing (n = 2)
What are and ?
0 0.5 1 1.5
2
0 0.5 1 1.5
2
0.75
0.75
Hypothesis Testing (n = 2)
What are and ?
0 0.5 1 1.5
2
0 0.5 1 1.5
2
0.75
0.75
= 1/8 = 0.125
Hypothesis Testing (n = 2)
What are and ?
0 0.5 1 1.5
2
0 0.5 1 1.5
2
0.75
0.75
= 1/8 = 0.125
= 1/8 = 0.125
Conclusion
By increasing the sample size, we can lower both and simultaneously.
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?
Example
The graph of the pdf of X3.
y0 1/3 2/3
3
1
Example
The graph of the pdf of X3.
y0 1/3 2/3 1
Area = 1
3
Example
What is the probability that X3 is between 1/3 and 2/3?
y0 1/3 2/3 1
3
Example
What is the probability that X3 is between 1/3 and 2/3?
y0 1/3 2/3 1
3
Example
The probability equals the area under the graph from 1/3 to 2/3.
y0 1/3 2/3 1
Area = 2/3
3
Verification
Use Avg3.xls to generate 10000 triples of numbers.
See if about 2/3 of the averages lie between 1/3 and 2/3.
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.
Hypothesis Testing (n = 3)
Distribution of X3 under H0:
Distribution of X3 under H1:0 1/3 2/3 1 4/3
0 1/3 2/3 1
1
4/3
Hypothesis Testing (n = 3)
Distribution of X3 under H0:
Distribution of X3 under H1:0 1/3 2/3 1 4/3
= 0.07
0 1/3 2/3 1
1
4/3
= 0.07
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?
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.
x1/2 2/31/3
N(1/2, 1/12)
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 0.4515.
Experiment
Use the Excel spreadsheet Avg12.xls to generate 10000 values of X, where X is the average of 12 random numbers from U(0, 1).
Test the 68-95-99.7 Rule.
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