1. Chap. 4 Continuous Distributions

2. Examples of Continuous Random Variable
If we randomly pick up a real number between 0 and 1, then we define a continuous uniform random variable V with , for any real number 0≤t≤1.

3. Distribution Function
The distribution function of a continuous random variable X is defined as same as that of a discrete random variable, i.e.

4. Probability Density Function
The probability density function（p.d.f.）of a continuous random variable is defined as
For example, the p.d.f. of an uniform random variable defined in [0,2] is

5. The p.d.f. of a continuous random variable with space S satisfies the following properties:

6. Uniform Distribution
Let random variable X correspond to randomly selecting a number in [a,b]. Then,

7. Uniform Distribution
X has a uniform distribution.

8. Some Important Observations
The p.d.f. of a continuous random variable does not have to be bounded. For example, the p.d.f. of a uniform random variable with space [0,1/m] is
The p.d.f. of a continuous random variable may not be continuous, as the above example demonstrates.

9. Expected Value and Variance
The expected value of a continuous random variable X is
The variance of X is

10. Expected Value of a Function of a Random Variable
Let X be a continuous random variable and Y=G(x). Then,
In the following, We will only present the proof for the cases, in which G(.) is a one-to-one monotonic function.

11. Expected Value of a Function of a Random Variable

12. Moment-Generating Function and Characteristic Function
The moment-generating function of a continuous random variable X is
Note that the moment-generating function, if it is finite for -h<t<h for some h>0, completely determines the distribution. In other words, if two continuous random variables have identical m.g.f., then they have identical probability distribution function.

13. Moment-Generating Function and Characteristic Function
The characteristics function of X is defined to be the Fourier transformation of its p.d.f.

14. Illustration of the Normal Distribution
Assume that we want to model the time taken to drive a car from the NTU main campus to the NTU hospital.
According to our experience, on average, it takes 20 minutes or 1200 seconds.

15. Illustration of the Normal Distribution
If we left the NTU main campus and drove to the NTU hospital now, then the probability that we would arrive in …seconds would be 0.
In addition, the probability that we would arrive in …seconds would also be 0.

16. Illustration of the Normal Distribution
However, our intuition tells us that it would be more likely that we would arrive within 600 seconds and 1800 seconds than within 3000 seconds and 4200 seconds.

17. Illustration of the Normal Distribution
Therefore, the likelyhood function of this experiment should be of the following form:

18. Illustration of the Normal Distribution
In fact, the probability density function models the likelihood of taking a specific amount of time to drive from the main campus to the hospital.
By the p.d.f., the probability that we would arrive within a time interval would be

19. Illustration of the Normal Distribution
In the real world, many distributions can be well modeled by the normal distributions. In other words, the profile of the p.d.f. of a normal distribution provides a good approximation of the exact p.d.f. of the distribution just like our example above.

20. The Standard Normal Distribution
A normal distribution with μ=0 and σ=1 is said to be a standard normal distribution.
The p.d.f. of a standard normal distribution is

21. Since is a circularly symmetric function on the Y-Z plane,
Therefore,c2=1 and c=1.

22. Linear Transformation of the Normal Distribution
Assume that random variable X has the distribution .
Then, has the standard normal distribution.
Proof：

23. Linear Transformation of the Normal Distribution
By substituting ,
we have
Therefore, Y is
Accordingly, if we want to compute , we can do that by the following procedure.

24. Expected Value and Variance of a Normal Distribution
Let X be a N(0,1).

25. Therefore, the expected value and variance of X are 0 and 1, respectively.
The expected value and variance of a
distribution are μ and σ2,
respectively,
since is N(0,1),
if Y is N(μ, σ2).

26. The Table for N（0,1）

27. The Chi-Square Distribution
Assume that X is N（0,1）.
In statistics, it is common that we are interested in
Therefore, we define Z=X2.
The distribution function of Z is

28. The Chi-Square Distribution
The p.d.f. of Z is

29. The Chi-Square Distribution
Z is typically said to have the chi-square distribution of 1 degree of freedom and denoted by

30. Chi-Square Distribution with High Degree of Freedom
Assume that X1, X2, ……, Xk are k independent random variables and each Xi is N(0, 1).
Then, the random variable Z defined by is called a chi-square random variable with degree of freedom = k and is denoted by

31. Addition of Chi-Square Distributions

32. Example of Chi-Square Distribution with Degree of Freedom = 2
Assume that a computer-controlled machine is commanded to drill a hole at coordinate (10,20). The machine moves the drill along the x-axis first followed by the y-axis.

33. Example of Chi-Square Distribution with Degree of Freedom = 2
According to the calibration process, the positioning accuracy of the machine in terms of millimeters along each axis is governed by a normal distribution N(u,0.0625).
The engineer in charge of quality assurance determines that the center of the hole must not deviate from the expected center by more than 1.0 millimeters.
What is the defect rate of this task.

34. The Table of the χ2 Distribution

35. The distribution Function and p.d.f. of is