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

EQT 272 PROBABILITY AND STATISTICS. ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS. ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS. Free Powerpoint Templates. CHAPTER 2. RANDOM VARIABLES.

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

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  1. EQT 272 PROBABILITY AND STATISTICS ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS 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 Example 2.3 can serve as the probability distribution of a discrete random variable. Solution

  11. 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

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

  13. Example 2.4: Consider the function • Find • Find

  14. Example 2.5 Let X be a continuous random variable with the following probability density function

  15. EXERCISE • A random variable x can assume 0,1,2,3,4. A probability distribution is shown here: (a) Check whether this is probability distribution. (b) Find (c) Find

  16. 2. Let 3. Let

  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 Solution

  20. Example 2.6 If X has the probability density

  21. 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

  22. 2.5.2 Variance

  23. 2.5.3 Standard Deviation 2.5.4 Properties of Expected Values • For any constant a and b,

  24. 2.5.5 Properties of Variances For any constant a and b,

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

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

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