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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)
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
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
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
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:
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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