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Chapter Two Random Variable L1- Discrete random variable L2- Continuous random variable
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Jan 2009 DISCRETE RANDOM VARIABLES <ul><li>At the end of the lecture, you will be able to : </li></ul><ul><li>describe types of random variables </li></ul><ul><li>calculate their probability distribution and their cumulative distribution </li></ul>Learning Objectives:
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Jan 2009 <ul><li>Random Variable : </li></ul><ul><li>A numerical variable whose measured value can change from is one outcome of random experiment. </li></ul><ul><li>An uppercase letter ( say X ) is used to denote a random variable. </li></ul><ul><li>After the experiment is conducted, the measured value is denoted by a lowercase letter, say x = 10. </li></ul><ul><li>Probability distribution / distribution of a random variable X : </li></ul><ul><li>description of the set of probabilities associated with the possible values of X . </li></ul><ul><li>Probability mass function: </li></ul><ul><li>describe the probability distribution of a discrete random variable </li></ul><ul><li>Probability density function: </li></ul><ul><li>describe the probability distribution of a continuous random variable </li></ul>
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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 }
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Jan 2009 DISCRETE RANDOM VARIABLE <ul><li>The set x of values of X is finite or countable. </li></ul>X is a discrete random variable if : <ul><li>The Probability Mass Function (pmf) of X is a set of probability values p i assigned to each of the values of x i </li></ul>
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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.
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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
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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.
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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
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Jan 2009 Graph of F( x ) F( x ) 50 200 350 0.5 0.3 1.0 Cost ( RM )
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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
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