Chap 6 Continuous Random Variables Ghahramani 3rd edition 2018/12/2
Outline 6.1 Probability density functions 6.2 Density function of a function of a random variable 6.3 Expectations and variances
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,
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.
Probability density functions
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?
Probability density functions Sol:
Probability density functions
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.235-237)
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.
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.
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.
Density function of a function of a random variable
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:
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: AR, 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
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
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
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.
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 .
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, .
Density function of a function of a random variable .
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 .
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
Density function of a function of a random variable
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
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.
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
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.
Expectations and variances Sol: (a) (b) To show that EX does not exist, note that
Expectations and variances Thm 6.2 For any continuous r. v. X with distribution function F and density function f,
Expectations and variances Proof:
Expectations and variances Thm 6.3 Let X be a continuous r. v. with density function f; then for any function h: R R,
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
Expectations and variances Ex 6.10 A point X is selected from the interval (0, ) randomly. Calculate E(cos2X) and E(cos2X). Sol:
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.
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.
Expectations and variances Sol: