This document discusses discrete and continuous random variables. It begins by defining a random variable as a numerical variable that can change value between outcomes of a random experiment. Discrete random variables take on countable values, while continuous variables can take any value within a range. The document provides examples of discrete random variables like the number of scratches on a surface. It then discusses probability mass functions for discrete variables and cumulative distribution functions, which are step functions representing the probabilities of values being below a given point. Formulas for mean, variance and standard deviation are also presented.

1. Chapter Two Random Variable L1- Discrete random variable L2- Continuous random variable

4. Jan 2009 Examples of random variables: The number of scratches on a surface. Integer values ranging from zero to about 5 are possible values. X = { 0, 1, 2, 3, 4,5} The time taken to complete an examination. Possible values are 15 minutes to over 3 hours. X = { 15 x 180 }

6. Jan 2009 Example 1: The sample space for a machine breakdown problem is S = { electrical, mechanical, misuse } and each of these failures is associated with a repair cost of about RM200, RM350 and RM50 respectively. Identify the random variable giving reasons for your answer. Example 2: The analysis of the surface of semi conductor wafer records the number of particles of contamination that exceed a certain size. Identify the possible random variable and its values.

7. Jan 2009 Probability mass function may typically be given in tabular or graphical form If from Example 1 that P( cost=50)=0.3, P (cost = 200) = 0.2 and P (cost = 350) = 0.5. The probability mass function is given either X = x 50 200 350 f ( x ) = P(X= x ) 0.3 0.2 0.5 tabular form line graph P( x ) 50 200 350 0.3 0.2 0.5 cost

8. Jan 2009 CUMULATIVE DISTRIBUTION FUNCTION The cumulative distribution F( x ) of a discrete random variable X with probability mass function f ( x ) is The cumulative distribution of F( x ) is an increasing step function with steps at the values taken by the random variable. The height of the steps are probabilities of taking these values.

9. Jan 2009 From Example 1 ( machine breakdowns) : The probability distribution is The following cumulative distribution is obtained X = x 50 200 350 f ( x ) = P(X= x ) 0.3 0.2 0.5

10. Jan 2009 Graph of F( x ) F( x ) 50 200 350 0.5 0.3 1.0 Cost ( RM )

11. Jan 2009 MEAN AND VARIANCE :- Discrete R.V We can summarize probability distribution by its mean and variance. Mean or expected value is Variance of X is given as Standard deviation of X is  