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Chapter 8 - ECMT1020 Chapter8 RandomVariable...

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ECMT1020 Chapter 8 Random Variable & Portfolio Analysis Dr Boris Choy for ECMT1020
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Topics covered 1. Mean and variance of discrete random variable 2. Mean & variance of a linear function of variable 3. Joint discrete probability distribution 4. Marginal and conditional distributions 5. Covariance and correlation coefficient 6. Portfolio analysis References Black 5.1, 5.2, 3.5 (p.93‐99)
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Learning Objectives Able to find the mean and variance of a discrete random variable and linear functions of the variable Understand the joint probability distribution and correlation between two discrete random variables. Using statistical method in portfolio analysis
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Discrete Random Variables
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Discrete Random Variables Definition: A random variable (r.v.) is a variable whose value is determined by the outcome of a random experiment, e.g. throw a dice or toss a coin. The r.v. that takes a finite number of values is a discrete random variable, e.g. the binomial random variable. The r.v. is denoted by X The value of the X from a specific experiment is x The probability of observing x is P( X = x ) or p(x)
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Dr Boris Choy ECMT1020: Chapter 8 6 Mean & Variance Definition: The mean and variance of a discrete r.v. X is defined by 2 possible all 2 2 2 2 possible all ) ( ] [ ] [ ] [ ) ( ] [ x x x p x X E X E X V x xp X E
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Dr Boris Choy ECMT1020: Chapter 8 7 Mean & Variance Example: x p ( x ) xp ( x ) x 2 p ( x ) 1.50 0.10 0.150 0.225 1.75 0.30 0.525 0.919 2.00 0.35 0.700 1.400 2.25 0.15 0.338 0.759 2.50 0.10 0.250 0.625 Sum 1.963 3.928 permitted) error (rounding 0767 . 0 9625 . 1 928125 . 3 ] [ ] [ ] [ 928125 . 3 ] [ and 9625 . 1 ) ( ] [ 2 2 2 2 X E X E X V X E x xp X E x
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Dr Boris Choy ECMT1020: Chapter 8 8 Mean & Variance Example: x p ( x ) xp ( x ) x 2 p ( x ) 0 0.2 0 0 1 0.5 0.5 0.5 2 0.2 0.4 0.8 3 0.1 0.3 0.9 Sum 1.2 2.2 _______ __________ __________ ] [ ] [ ] [ _ __________ ] [ and _________ ) ( ] [ _ __________ ) 2 1 ( _____ __________ ) 2 ( 2 2 2 X E X E X V X E x xp X E X P X P x
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Dr Boris Choy ECMT1020: Chapter 8 9 Mean of a Linear Function of X Consider Y = a + bX E[Y] = E[ a + bX ] = a + b E[ X ] Proof:
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Dr Boris Choy ECMT1020: Chapter 8 10 Variance of a Linear Function of X V[ Y ] = V[ a + bX ] = b 2 V[ X ] Proof:
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Dr Boris Choy ECMT1020: Chapter 8 11 Mean & Variance of a Linear Function of X Exercises: __________ __________ ] 3 [ (f) __________ __________ ] 3 2 [ (e) _ __________ __________ ] 1 [ (d) __________ __________ ] 4 [ (c) __________ __________ ] 2 [ (b) __________ __________ ] 1 3 [ (a) ] [ : NB . ] [ and ] [ Let 2 1 2 2 1 2 2 2 2 X V X V X V X E X E X E X E X V X E
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Dr Boris Choy ECMT1020: Chapter 8 12 Example Example: (Binomial distribution) An auditor is preparing for a physical count of inventory as a means of verifying its value.
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