Bivariate Random Variables

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Speci cally, P(Y = 1) = P(X = 1 and Y = 1) + P(X = 2 and Y = 1) + P(X = 3 and Y = 1) = 1/12 + 1/12 + 1/3 = 1/2 In a similar way, we nd P(Y = 2) = 1/2 We then found the following probability distributions for the individual variables: 5/12 if x = 1 if x = 2 f (x) = 1/6 5/12 if x = 3 and g(y) = 1/2 1/2 if y = 1 if y = 2

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These distributions occur in the margins of the table and are called marginal distributions We have expanded Table 81 to show these marginal distributions in Table 82

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Table 82 X Y 1 2 f (x) 1 1/12 1/3 5/12 2 1/12 1/12 1/6 3 1/3 1/12 5/12 g(y) 1/2 1/2 1

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Note that where the sums are over all the values of X and Y, P(X = x, Y = y) = P(Y = y) = g(y)

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and P(X = x, Y = y) = P(X = x) = f (x)

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These random variables also have expected values We nd E(X) = 1 5 1 5 +2 +3 =2 12 6 12 1 1 3 E(Y ) = 1 + 2 = 2 2 2

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Now we note that E(X + Y ) =

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= E(X) + E(Y )

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8

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Continuous Probability Distributions

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This is not a peculiarity of this special case, but is, in fact, true for any two variables X and Y Here is a proof E(X + Y ) =

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(x + y)P(X = x, Y = y) xP(X = x, Y = y) +

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x y x y

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yP(X = x, Y = y) y

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P(X = x, Y = y) +

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P(X = x, Y = y)

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xf (x) +

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y g(y) = E(X) + E(Y )

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This is easily extended to any number of random variables: E(X + Y + Z + ) = E(X) + E(Y ) + E(Z) + When more than one random variable is de ned on the same sample space, they may be related in several ways: they may be totally dependent as, for example, if X = Y or if X = Y 4, they may be totally independent of each other, or they may be partially dependent on each other In the latter case, the variables are called correlated This will be dealt with when we consider the subject of regression later It is important to emphasize that E(X + Y + Z + ) = E(X) + E(Y ) + E(Z) + no matter what the relationships are between the several variables, since no condition was used in the proof above Note in the example we have been considering that P(X = 1 and Y = 1) = 1/12 = P(X = 1) P(Y = 1) = 5/12 1/2 = 5/24 / so X and Y are not independent Now consider an example where X and Y are independent of each other We show another joint probability distribution function in Table 83

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Table 83 X y 1 2 f (x) 1 5/24 5/24 5/12 2 1/12 1/12 1/6 3 5/24 5/24 5/12 g(y) 1/2 1/2 1

Bivariate Random Variables

Note that P(X = x, Y = y) = P(X = x)P(Y = y) in each case, so the random variables are independent We can calculate, for example, P(X = 1 and Y = 1) = 5/24 = P(X = 1) P(Y = 1) = 1/12 1/2 The other entries in the table can be checked similarly Here we have shown the marginal distributions of the random variables X and Y , f (x) and g(y), in the margins of the table Now consider the random variable X Y and in particular its expected value Using the fact that X and Y are independent, we sum the values of X Y multiplied by their probabilities to nd the expected value of the product of X and Y : E(X Y ) = 1 1 1 2 5 1 1 5 +1 2 +1 3 12 6 2 12 1 1 5 1 +2 3 6 2 12 2

1 5 +2 1 2 12

+2 2 =3