1. Chap 6 Continuous Random Variables Ghahramani 3rd edition
2018/12/2

2. Outline 6.1 Probability density functions
6.2 Density function of a function of a
random variable
6.3 Expectations and variances

3. 6.1 Probability density functions
Def Let X be a random variable. Suppose that there exists a non-negative real-valued function f: R[0, ) such that for any subset of real numbers A which can be constructed from intervals by a countable number of set operations,

4. 6.1 Probability density functions
Then X is called absolutely continuous or in this book, for simplicity, continuous. The function f is called the probability density function, or simply, the density function of X.
Let f be the density function of a random variable X with distribution function F. Some immediate properties of f are as follows.

5. Probability density functions

6. Probability density functions
Ex 6.1 Experience has shown that while walking in a certain park, the time X, in minutes, between seeing 2 people smoking has a density function of the form
(a) Calculate the value of .
(b) Find the probability distribution function of X.
(c) What is the probability that Jeff, who has just seen a person smoking, will see another person smoking in 2 to 5 minutes? In at least 7 minutes?

7. Probability density functions
Sol:

8. Probability density functions

9. Probability density functions
Ex 6.2
(a) Sketch the graph of the function
and show that it is the probability density
function of a random variable X.
(b) Find F, the distribution function of X, and show that it is continuous.
(c) Sketch the graph of F.
Sol: (see pp )

10. 6.2 Density function of a function of a random variable
f is the density function of a random variable X, then what is the density function of h(X)?
2 methods for it:
(a) the method of distributions: to find the density of h(X) by calculating its distribution function;
(b) the method of transformations: to find the density of h(X) directly from the density function of X.

11. Density function of a function of a random variable
(Method of distributions)
Ex 6.3 Let X be a continuous random variable with the probability density function
Find the distribution and density functions of Y=X2.

12. Density function of a function of a random variable
Sol: Let G and g be the distribution function and the density function of Y, respectively.

13. Density function of a function of a random variable

14. Density function of a function of a random variable
Ex 6.4 Let X be a continuous random variable with distribution function F and probability density function f. In terms of f, find the distribution and the density functions of Y=X3.
Sol:

15. Density function of a function of a random variable
Thm 6.1 (Method of Transformations) Let X be a continuous random variable with density function fX and the set of possible values A. For the invertible function h: AR, let Y=h(X) be a random variable with the set of possible values B=h(A)={h(a): a in A}. Suppose that the inverse of y=h(x) is the function x=h-1(y), which is differentiable for all values of y in B. Then fY, the density function of Y, is given by

16. Density function of a function of a random variable
Proof: Let FX and FY be distribution functions of X and Y=h(X), respectively. Differentiability of h-1 implies that it is continuous. Since a continuous invertible function is strictly monotone, h-1 is either strictly increasing or strictly decreasing. If it is strictly increasing,
and hence

17. Density function of a function of a random variable
Moreover, in this case h is also strictly increasing, so
Differentiating this by chain rule,
which gives the theorem. If h-1 is strictly decreasing,
and hence

18. Density function of a function of a random variable
In this case h is also strictly decreasing
Differentiating this by chain rule,
Showing that the theorem is valid in this case as well.

19. Density function of a function of a random variable
Ex 6.6 Let X be a random variable with the density function
Using the method of transformation, find the density function of Y=X1/2 .

20. Density function of a function of a random variable
Sol: The set of possible values of X is A=(0, ). Let h: (0, )R be defined by h(x)=x1/2. We want to find the density function of Y= h(X)=X1/2. The set of possible values of h(X) is B={h(a): a in A}=(0, ). The function h is invertible with the inverse x=h-1(y)=y2, which is differentiable, and its derivative is
(h-1)’(y)=2y. Therefore by Thm 6.1, .

21. Density function of a function of a random variable.

22. Density function of a function of a random variable
Ex 6.7 Let X be a continuous random variable with the density function
Using the method of transformation, find the density function of Y=1-3X2 .

23. Density function of a function of a random variable
Sol: The set of possible values of X is
A=(0,1). Let h: (0,1)R be defined by h(x)=1-3x2. The set of possible values of h(X) is B={h(a): a in A}=(-2,1). Since the domain of the function h is (0,1), h is invertible, and its inverse is found by solving 1-3x2=y for x, which gives

24. Density function of a function of a random variable

25. 6.3 Expectations and variances
Def If X is a continuous r. v. with density function f, the expected value of X is defined by

26. Expectations and variances
Ex 6.8 In a group of adult males, the difference between the uric acid value and 6, the standard value, is a random variable X with the following density function:
Calculate the mean of these differences for the group.

27. Expectations and variances
Sol:
Remark: If X is a continuous r. v. with density function f, X is said to have a finite expected value if

28. Expectations and variances
Ex 6.9 A r. v. X with density function
is called a Cauchy random variable.
(a) Find c.
(b) Show that EX does not exist.

29. Expectations and variances
Sol: (a)
(b) To show that EX does not exist, note that

30. Expectations and variances
Thm 6.2 For any continuous r. v. X with distribution function F and density function f,

31. Expectations and variances
Proof:

32. Expectations and variances
Thm 6.3 Let X be a continuous r. v. with density function f; then for any function
h: R  R,

33. Expectations and variances
Coro Let X be a continuous r. v. with density function f(x). Let h1, h2, …, hn be real-valued functions, and let a1, a2, …, an be real numbers. Then

34. Expectations and variances
Ex 6.10 A point X is selected from the interval (0, ) randomly. Calculate
E(cos2X) and E(cos2X).
Sol:

35. Expectations and variances
Def Var(X) for a continuous r. v. is similar to that for a discrete r. v..
Var(X) = E(X2) - (EX)2
The moments, absolute moments, moments about a constant c, and central moments are all defined similarly.

36. Expectations and variances
Ex 6.11 The time elapsed, in minutes, between the placement of an order of pizza and its delivery is random with the density function
(a) Determine the mean and standard deviation of the time it takes for the pizza shop to deliver pizza.
(b) Suppose that it takes 12 minutes for the pizza shop to bake pizza. Determine the mean and standard deviation of the time it takes for the delivery person to deliver pizza.

37. Expectations and variances
Sol: