CHAPTER 4 INTRODUCTION TO RANDOM VARIABLE CHAPTER OUTLINE

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CHAPTER 4 INTRODUCTION TO RANDOM VARIABLE CHAPTER OUTLINE: DISCRETE RANDOM VARIABLE v CONTINUOUS RANDOM VARIABLE v MEAN & VARIANCE OF RANDOM VARIABLE v

Introduction to Random Variable Introduction Every trial / experiments, has one or more possible outcomes. Outcomes occur randomly. We often summarize the outcome from a random experiment by a simple number. Definition 4. 1 A variable is a symbol such as X or Y 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.

Example 4. 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 (Number of getting head) HH ¼ 2 HT ¼ 1 TH ¼ 1 TT ¼ 0 Also we can write P(X=2)=1/4, for example, for the probability of the event that the random variable X will take on the value 2.

Two Types of Random Variable Definition 4. 2 1) Discrete Random Variables A random variable is discrete if its set of possible values consist of discrete points / finite number on the number line. Assume values that are countable (0, 1, 2, 3). Example: • The number of cars sold at a dealership during a month • The number of customers who visit a bank • The number of houses in a certain block

2) Continuous Random Variables A random variable is continuous if its set of possible values consist of an entire interval on the number line / infinite values. Continuous random variables are usually measurements. Examples: • The height of a person • The time taken to complete an examination • The weight of a fish • The price of a house

Discrete Random Variable 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 4. 2 Check whether the distribution is a probability distribution. X 0 1 2 3 4 P(X=x) 0. 125 0. 375 0. 025 0. 375 0. 125 Solution: # so the distribution is not a probability distribution.

Example 4. 3 Check whether the function given by, can serve as the probability distribution of a discrete random variable.

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

Cumulative Distribution Function of Discrete Random Variable The cumulative distribution function F(x), of discrete random variable X is denoted as: For a discrete random variable X, F(x) satisfies the following properties: If the range of a random variable X consists of the values

Example 4. 4 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

Continuous Random Variable 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:

Example 4. 5 Let X be a continuous random variable with the following probability density function,

Solution: 1)

Cumulative Distribution Function of Continuous Random Variable The cumulative distribution function of a continuous random variable X:

Example 4. 6 If X has the probability density,

Solution:

Mean, Variance & Std. Deviation of Random Variable 1) Mean 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,