In Example 34 we have f Y y Z 5 f XY x y dx Z 5 2 xe y dx e y y In Example 35

In example 34 we have f y y z 5 f xy x y dx z 5 2 xe

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In Example 3.4, we have f Y ( y ) = Z 0 . 5 0 f X,Y ( x, y ) dx = Z 0 . 5 0 2 xe - y dx = e - y , y > 0 . In Example 3.5, we have f Y ( y ) = f X,Y (1 , y ) + f X,Y ( - 1 , y ) = 0 . 5 2 π e - 1 2 ( y - 1) 2 + e - 1 2 ( y +1) 2 . 4
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Sometimes, we are interested in the conditional random variable X | Y = y , e.g. the stock price given good economy. This is in a sense that we are “living in the world of Y = y ”. Therefore, our universe is (the sample space of) Y = y and we should (zoom into the event Y = y and) rescale our probability such that P (All possibilities | Y = y ) = 1. In view of this, we define the conditional probability function f X | Y ( x | y ) = f X,Y ( x, y ) f Y ( y ) . Note the similarity between this and the conditional probability for events P ( A | B ) = P ( A B ) P ( B ) provided that P ( B ) 6 = 0. 3.4. Descriptive Statistics We often want to assess the random variable (a function F X ) using a few numbers. The most common ones are the moments, especially the first 2 moments: the mean (expected value) gives roughly the “location” and variance (related to the second moment) gives the “spread”. For discrete random variables, the formula is E ( X ) = X x Ω xf X ( x ) . Example 3.8. For X Binomial (2 , 0 . 3) , we have f X (0) = 0 . 49 , f X (1) = 0 . 42 , f X (2) = 0 . 09 . Therefore, the expected value of X is given by E ( X ) = 0 × 0 . 49 + 1 × 0 . 42 + 2 × 0 . 09 = 0 . 42 + 0 . 18 = 0 . 6 . For continuous random variables, the formula is E ( X ) = Z x Ω xf X ( x ) dx = Z x Ω xdF X ( x ) . Example 3.9. For X exp( λ = 2) , we have f X ( x ) = 2 e - 2 x , x > 0 . Its expected value is given by E ( X ) = Z 0 x 2 e - 2 x dx = Z 0 - xd ( e - 2 x ) ( Integration by parts ) = [ - xe - 2 x ] x →∞ x =0 + Z 0 e - 2 x dx = 1 2 Z 0 2 e - 2 x dx = 1 2 . 5
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If furthermore X 0, then E ( x ) = Z x Ω (1 - F X ( x )) dx where (1 - F X ( x )) is sometimes called the survival function, denoted by either F X or S X . Example 3.10. In the above example, we have 1 - F X ( x ) = e - 2 x and therefore the expected value can be calculated as E ( X ) = Z 0 e - 2 x dx = 1 2 Z 0 2 e - 2 x dx = 1 2 . For multiple random variables, we have E ( X | Y = y ) = ( x :( x,y ) Ω xf X | Y ( x | y ) if X is discrete R x :( x,y ) Ω xf X | Y ( x | y ) dx if X is continuous which is a function of y . Note if we do not assign a value to Y , the above E ( X | Y ) would be a function of Y , i.e. a random variable, as we do not know what Y is. Hence, E ( X | Y ) is a random variable. Its mean is E [ X ], i.e. E ( E ( X | Y )) = E [ X ] . (Double expectation formula / Tower property) This implies the conditional variance formula: V ar ( X ) = E ( V ar ( X | Y )) + V ar ( E ( X | Y )) . 3.5. Expectation and Related Quantities Note from the formula of E that E is linear, i.e. for a real number a and 2 random variables X and Y , we have E ( aX ) = a E ( X ) , and E ( X + Y ) = E ( X ) + E ( Y ) . Variance is defined as V ar ( X ) = E (( X - E ( X )) 2 ) = E ( X 2 ) - ( E ( X )) 2 which captures the dispersion of the random variable X . Covariance between 2 random variables X and Y is defined as Cov ( X, Y ) = E (( X - E ( X ))( Y - E ( Y ))) = E ( XY ) - E ( X ) E ( Y ) which measures the degree of (linear) dependence of them. In particular, the Pearson correlation defined as ρ ( X, Y ) = Cov ( X, Y ) p V ar ( X ) V ar ( Y ) removes the dependencies on the units of X and Y . The Pearson correlation can only take values between - 1 and 1. It is - 1 when Y = α + βX for some real number α and negative number β , i.e. a linear relationship.
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