{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

chap05PRN econ 325

50 a f 50 a z p 50 380 a 380 x p a 380 x p the

Info iconThis preview shows pages 15–18. Sign up to view the full content.

View Full Document Right Arrow Icon
< 50 a F 50 a Z P 50 380 ) a 380 ( X P ) a 380 X ( P The Appendix Table gives 90 . 0 ) ( F = 1.28 Therefore, 1.28 = 50 a and 64 50 1.28 = = ) )( ( a The range centered at $380 is: [ $380 – 64, $380 + 64] = [ $316, $444] As a check on the calculations, the upper limit can be calculated with Microsoft Excel by using the function: NORMINV(0.9, 380, 50) Econ 325 – Chapter 5 30 Chapter 5.6 Jointly Distributed Continuous Random Variables Results stated earlier for jointly distributed discrete random variables can be extended to work with continuous random variables. Let X and Y be two continuous random variables that take numeric values denoted by x and y , respectively. The joint cumulative distribution function (CDF) is: ) y Y x X ( P ) y , x ( F Y , X < < = and The marginal distribution functions are: ) x X ( P ) x ( F X < = and ) y Y ( P ) y ( F Y < = X and Y are statistically independent if and only if: ) y ( F ) x ( F ) y , x ( F Y X Y , X =
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Econ 325 – Chapter 5 31 A measure of linear association is covariance: Y X Y X ) XY ( E )] Y ( ) X [( E ) Y , X ( Cov μ μ - = μ - μ - = where ) X ( E X = μ and ) Y ( E Y = μ If X and Y are independent then 0 = ) Y , X ( Cov . However, zero covariance does not guarantee independence. X and Y may have some complicated non-linear relationship. h Special Case: If X and Y are joint normally distributed random variables then zero covariance also gives the result that X and Y are independent. Econ 325 – Chapter 5 32 x Linear Combinations of Random Variables For constant fixed numbers a and b , a linear combination of random variables X and Y is: Y b X a W + = The mean of the random variable W is: ) Y ( E b ) X ( E a ) W ( E W + = = μ The variance of W is: ) Y , X ( Cov b a 2 ) Y ( Var b ) X ( Var a ) W ( Var σ 2 2 2 W + + = = h Special Case: If X and Y are joint normally distributed random variables then Y b X a W + = is also normally distributed with mean and variance as given above. That is, ) σ , ( N ~ W 2 W W μ
Background image of page 16
Econ 325 – Chapter 5 33 Now consider three random variables 1 X , 2 X and 3 X with means 1 μ , 2 μ and 3 μ and variances 2 1 σ , 2 σ 2 and 2 σ 3 . The sum of these random variables has the properties: 3 2 3 2 ) X X X ( E μ + μ + μ = + + 1 1 and ) X , X ( Cov 2 ) X , X ( Cov 2 ) X , X ( Cov 2 ) X X X ( Var 3 2 3 1 2 1 3 2 1 3 2 + + + σ + σ + σ = + + 2 2 2 1 With independence the covariance between every pair of these random variables is zero to give a simpler result for the variance of the sum: 2 2 2 1 σ + σ + σ = + + 3 2 1 3 2 ) X X X ( Var Econ 325 – Chapter 5 34 Example: Portfolio Analysis The random variables X and Y are the share prices of two companies trading on the stock market such that ) , ( N ~ X 81 25 and ) , ( N ~ Y 121 40 X μ 2 X σ Y μ 2 Y σ The correlation between the two stock prices is: 0.4 - = ρ XY A portfolio is the random variable: Y X W 30 20 + = Find the probability that the portfolio value exceeds 2,000.
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 18
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page15 / 18

50 a F 50 a Z P 50 380 a 380 X P a 380 X P The Appendix...

This preview shows document pages 15 - 18. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online