ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS

1. 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

2. CHAPTER 2 RANDOM VARIABLES

3. 2. RANDOM VARIABLES

4. 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 variableis a variable whose value is determined by the outcome of a random experiment.

5. 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

6. TWO TYPES OF RANDOM VARIABLES

7. EXAMPLES

8. 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:

9. Check whether the distribution is a probability distribution. • so the distribution is not a probability distribution. Example 2.2 Solution

10. 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 Example 2.3 can serve as the probability distribution of a discrete random variable. Solution

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

12. 2.3 CONTINUOUS PROBABILITY DISTRIBUTIONS Definition 2.4: • A function with valuesf(x),defined over the set of all numbers, is called a probability density function of the continuous random variable X if and only if

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

14. Example 2.4 Let X be a continuous random variable with the following probability density function

15. Solution a)

16. b)

17. 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

18. The cumulative distribution function of a continuous random variable X is

19. Example 2.5 (Discrete random variable) Solution

20. 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

21. Solution For , For

22. For ,

24. Example 2.6 b.(Continuous random variable) If X has the probability density

25. Solution

28. 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

29. 2.5.2 Variance

30. 2.5.3 Standard Deviation 2.5.4 Properties of Expected Values • For any constant a and b,

31. 2.5.5 Properties of Variances For any constant a and b,

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

33. Solution Mean:

34. Varians:

35. Example 2.8 Let X be a continuous random variable with the Following probability density function

36. Solution