EQT 272 PROBABILITY AND STATISTICS ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS MADAM ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS Free Powerpoint Templates

CHAPTER 2 RANDOM VARIABLES

2. RANDOM VARIABLES 2.1Introduction 2.2Discrete probability distribution 2.3Continuous probability distribution 2.4Cumulative distribution function 2.5Expected value, variance and standard deviation

INTRODUCTION In an experiment of chance, outcomes occur randomly. We often summarize the outcome from a random experiment by a simple number. Definition 2.1 A variable is a symbol such as X, Y, Z, x or H, that assumes values for different elements. If the variable can assume only one value, it is called a constant. A random variable is a variable whose value is determined by the outcome of a random experiment.

Elements of sample space Example 2.1 A balanced coin is tossed two times. List the elements of the sample space, the corresponding probabilities and the corresponding values X, where X is the number of getting head. Solution Elements of sample space Probability X HH ¼ 2 HT 1 TH TT

Discrete Random Variables Continuous Random Variables TWO TYPES OF RANDOM VARIABLES A random variable is discrete if its set of possible values consist of discrete points on the number line. Discrete Random Variables A random variable is continuous if its set of possible values consist of an entire interval on the number line. Continuous Random Variables

EXAMPLES Examples of continuous random variables: Examples of discrete random variables: -number of scratches on a surface -proportion of defective parts among 1000 tested -number of transmitted bits received error Examples of continuous random variables: -electrical current -length -time

2.2 DISCRETE PROBABILITY DISTRIBUTIONS Definition 2.3: If X is a discrete random variable, the function given by f(x)=P(X=x) for each x within the range of X is called the probability distribution of X. Requirements for a discrete probability distribution:

Example 2.2 Solution Check whether the distribution is a probability distribution. All the probabilities between 0 and 1. . so the distribution is not a probability distribution. x 1 2 3 4 P(X=x) 0.125 0.375 0.025

Example 2.3 Check whether the function given by can serve as the probability distribution of a discrete random variable. Solution Check whether the function given by The probabilities are between 0 and 1 f(1) = 3/25 , f(2) = 4/25, f(3) =5/25, f(4) = 6/25, f(5)=7/25

Solution # so the given function is a probability distribution of a discrete random variable.

2.3 CONTINUOUS PROBABILITY DISTRIBUTIONS Definition 2.4: A function with values f(x), defined over the set of all numbers, is called a probability density function of the continuous random variable X if and only if

Requirements for a probability density function of a continuous random variable X:

Example 2.4 Let X be a continuous random variable with the following Verify whether this distribution is a probability density function Find

Answer; The distribution is probability density function if it fulfill the following requirements, All f(x)≥0 If In this problem, First requirement f(0)=3/4≥0, f(1)=3/2≥0, f(x)=0, otherwise - All f(x)≥0 Must write the conclusion so that we know the first requirement is fulfill!

Write last conclusion to answer the question! 2) Second requirement - Since all requirements all fulfill, the distribution is probability density function. Must write the conclusion so that we know the second requirement is fulfill! Write last conclusion to answer the question!

2.4 CUMULATIVE DISTRIBUTION FUNCTION The cumulative distribution function of a discrete random variable X , denoted as F(X), is For a discrete random variable X, F(x) satisfies the following properties: If the range of a random variable X consists of the values

The cumulative distribution function of a continuous random variable X is

Example 2.5 (Discrete random variable) Solution x 1 2 3 4 f(x) 4/10 3/10 2/10 1/10 F(x) 4/10 7/10 9/10 1

Example 2.6 a.(Continuous random variable) Given the probability density function of a random variable X as follows; Find the cumulative distribution function, F(X) Find

Solution i) Cummulative distribution function, F(X) For , For

For , Summary is important!!!

Two methods to answer question 3

TRY!!! The lifetime of a computer, in x years, is given by a random variable with the following function. Verify that the function is probability density function. Find the cumulative distribution function of the random variable X. Find P(1≤X≤3)

If X is a discrete random variable, 2.5 EXPECTED VALUE, VARIANCE AND STANDARD DEVIATION 2.5.1 Expected Value The mean of a random variable X is also known as the expected value of X as If X is a discrete random variable, If X is a continuous random variable,

2.5.2 Variance

2.5.4 Properties of Expected Values 2.5.3 Standard Deviation 2.5.4 Properties of Expected Values For any constant a and b,

2.5.5 Properties of Variances For any constant a and b,

Example 2.7 Find the mean, variance and standard deviation of the probability function

Solution Mean:

Varians:

Example 2.8 Let X be a continuous random variable with the following probability density function

Answer; a) E (X) =

b) Var (X) = E (X2) – (E(X))2 E (X) = 1,

Exercise 1 A survey asked a sample of people living in Kangar, how many times they shop at The Store supermarket each week and the following distribution is obtained; Does the distribution determine the probability distribution for discrete random variable? Find the probability of people shop at The Store more than 2 times. (answer: 0.2) Find the mean and variance of number of times people living in Kangar shop at The store supermarket. (mean=1.54, variance=1.4084) X 1 2 3 4 P(X=x) 0.2 0.35 0.25 0.11 0.09

Exercise 2 A restaurant manager is considering a new location for her restaurant. The projected annual cash flow for the new location is; Given that the distribution is probability distribution, find the expected cash flow for the new location. (E(X)=64 000) Annual cash flow $10,000 $30,000 $70,000 $90,000 $100,000 Probability 0.10 0.15 0.50 ?????

Exercise 3 X is a random variable which represents the delay time, in minutes, for a Star Cruise to depart from Pulau Langkawi jetty, with function Verify that the function is probability density function. Calculate F (X). Find E(X) and Var (X). Find P(10≤X≤30)