Learning Outcomes: Probability density function is usually denoted by (i) probability density function. [Include: Mode.] (ii) cumulative distribution function. [Include: Graph sketching & median.] (a) Find the Find the probabilities from probability density function and cumulative distribution function. [Restrict to polynomials, and only] (b) At the end of the lesson, students should be able to Find the probability density function from a cumulative distribution function and vice versa. (c) Compute the expectation and variance of continuous random variables. [Include: Properties] (d) Continuous Random Variables is a continuous random variable with the probability density function, (I) for all values valid in its interval if satisfy both the conditions below: the probability distribution function corresponding to a continuous random variable and denoted by Probability Density Function Properties: [ are real number constants] • • • • •If is the median of , then • Probability = Area under the graph of 8.2 Continuous Random Variables C08 -SM025 Page 1
Show that the function is the probability density function of a random variable X. a) Given for . Solution: Since for and , is the probability density function of X. [Shown] b) Find . Solution: Note that Note that Example 1 (a) (b) C08 -SM025 Page 2
c) Find the median, . Given for . Solution: Since , d) Sketch the graph of probability density function. Hence, state the mode. Solution: Note that Example 1 (c) (d) C08 -SM025 Page 3
A continuous random variable has probability density function a) Find the value of for . Solution: b) Find . Solution: c) Find . Solution: Note that Example 2 (a) (b) C08 -SM025 Page 4
A continuous random variable has probability density function. a) Find the value of the constant . Solution: b) Sketch the graph of probability density function Solution: Example 3 (a) (b) C08 -SM025 Page 5
c) Find and Solution: Example 3 (c) C08 -SM025 Page 6
d) Find the median. Solution: Let [to 3 significant figures] Since , Example 3 (d) C08 -SM025 Page 7
If is a continuous random variable with the p.d.f. , , then the cumulative distribution function of , is given by •The integral of is written in term of to avoid confusion in calculation to get • If is the median of , then find based on [ divides the area under the graph of p.d.f. into 2 equal parts] Remarks: Continuous Cumulative Distribution C08 -SM025 Page 8
is a continuous random variable with probability density function. Solution: For For For Example 1 (a) C08 -SM025 Page 9
Solution: Solution: Let [to 3 significant figures] Since , Example 1 (b) (c) C08 -SM025 Page 10
A continuous random variable has probability density function. a) Obtain the cumulative distribution function . Solution: Example 2 (a) C08 -SM025 Page 11
b) Sketch the graph of . Solution: c) Find and Solution: Example 2 (b) (c) C08 -SM025 Page 12
d) Determine the median, . Solution: So, . Example 2 (d) C08 -SM025 Page 13
The cumulative distribution function, of a continuous random variable is defined as a) Find the probability density function . Solution: b) Hence, find . Solution: Example 3 C08 -SM025 Page 14
The cumulative distribution function, of a continuous random variable is defined as a) Find the value of and . Solution: In the graph of In the graph of Example 4 (a) *[Exercise in Lecture 4 of 5 in PowerPoint] C08 -SM025 Page 15
b) Find the probability density function . Solution: Example 4 (b) *[Exercise in Lecture 4 of 5 in PowerPoint] C08 -SM025 Page 16
(a) The cumulative distribution function, of a continuous random variable is defined as . Based on given, find Solution: (b) Solution: (c) . Solution: Example 5 *[Extra Question] C08 -SM025 Page 17
Consider a continuous random variable with the p.d.f. . • also known as Expected Value of or Expectation of • denoted by or • given by ▪ ▪ ▪ ▪ Properties: [ are real number constants] Mean of denoted by or • • given by ▪ ▪ Properties: [ are real number constants] Variance of • denoted by • given by Standard Deviation of Mean & Variance (Continuous Random Variable) C08 -SM025 Page 18
A continuous random variable has the following probability density function. Solution: Example 1 (a)(b) Example 2 in Lecture 5 of 5 in PowerPoint] C08 -SM025 Page 19
Solution: Example 1 (b) *[Example 2 in Lecture 5 of 5 in PowerPoint] C08 -SM025 Page 20
Solution: Example 1 (c) *[Example 1 in Lecture 5 of 5 in PowerPoint] C08 -SM025 Page 21
A continuous random variable has the following probability density function. Solution: Solution: Example 2 (a) (b) *[Example 1 in Lecture 5 of 5 in PowerPoint] C08 -SM025 Page 22
Solution: Solution: Example 2 (c) (d) C08 -SM025 Page 23
Solution: Example 3 *[Exercise in Lecture 5 of 5 in PowerPoint] C08 -SM025 Page 24