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ch1pt2 - Chapter 1 Review of Probability Random Variable...

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Chapter 1: Review of Probability & Random Variable Concepts Random Variable Dr. Lim HS Last Updated: 21 May 2009 c 2009 MMU

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Presentation outline The random variable concept Distribution and density functions Examples of distribution and density functions Conditional distribution and density functions Expectation and Moments Transformations of a Random Variable Computer Generation of a Random Variable * c 2009 MMU Page 1/45
The random variable concept Random Variable ƒ A random variable (rv) is a function that maps each point (or outcome) in sample space S into some point on the real line. ƒ A rv is denoted by capital letter (e.g., W , X , or Y ). ƒ A particular value of the rv is denoted by lowercase letter (e.g., w , x , or y ). ƒ We assume that the mapping is unique, i.e., every point in S must correspond to only one value of the rv. c 2009 MMU Page 2/45

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The random variable concept Random Variable (cont.) Example 1. An experiment consists of rolling a die and flipping a coin. The applicable sample space is illustrated in Figure 1. Let the random variable be a function X chosen such that - a coin head ( H ) outcome corresponds to positive values of X that are equal to the numbers that show up on the die. - a coin tail ( T ) outcome corresponds to negative values of X that are equal in magnitude to twice the number that shows on the die. c 2009 MMU Page 3/45
The random variable concept Random Variable (cont.) Figure 1: A random variable mapping of a sample space. c 2009 MMU Page 4/45

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The random variable concept Random Variable (cont.) ƒ Rv may be grouped into 3 types: - Discrete rv : one that may take on only discrete values; - Continuous rv : one having a continuous range of values; - Mixed rv : one with both discrete and continuous values. c 2009 MMU Page 5/45
Distribution and Density Functions Distribution Function ƒ The distribution function , also called the cumulative distribution function (cdf), of a rv X is defined as F X ( x ) = P { X x } where -∞ < x < . ƒ P { X x } is therefore the probability of the event { X x } . c 2009 MMU Page 6/45

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Distribution and Density Functions Distribution Function (cont.) Example 2. Let X ∈ {- 1 , - 0 . 5 , 0 . 7 , 1 . 5 , 4 } . The corresponding probabilities are assumed to be { 0 . 1 , 0 . 2 , 0 . 1 , 0 . 4 , 0 . 2 } . The distribution function is shown in Figure 2. Figure 2: Distribution function applicable to the discrete rv. c 2009 MMU Page 7/45
Distribution and Density Functions Distribution Function (cont.) Example 3. Let the fair wheel-of- chance be numbered from 0 to 12. Clearly, P { X 0 } = 0 . For 0 < x 12 , P { X x } will increase linearly with x . The distribution function is shown in Figure 4. spin 12 3 6 9 Figure 3: The fair wheel-of-chance. 0 12 x F X ( x ) 1 0.5 6 Figure 4: Distribution function applicable to the continuous rv. c 2009 MMU Page 8/45

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Distribution and Density Functions Distribution Function (cont.) ƒ Properties of distribution function: 1. F X ( -∞ ) = 0 2. F X ( ) = 1 3. 0 F X ( x ) 1 4. F X ( x 1 ) F X ( x 2 ) if x 1 < x 2 5. P { x 1 < X x 2 } = F X ( x 2 ) - F X ( x 1 ) 6. F X ( x + ) = F X ( x ) c 2009 MMU Page 9/45
Distribution and Density Functions Density Function ƒ The

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ch1pt2 - Chapter 1 Review of Probability Random Variable...

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