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STATLECT > Additional topics in probability theory > Joint moment generating function

Joint moment generating function - Exercise set 1

This exercise set contains some solved exercises on joint moment generating functions. The theory needed to solve these exercises is introduced in the lecture entitled Moment generating function of a random vector.

Exercise 1.1

Let X be a $2 imes 1$ discrete random vector and denote its components by X_1 and X_2. Let the support of X be: [eq1]and its joint probability mass function be:[eq2]Derive the joint moment generating function of X, if it exists.

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Using the definition of moment generating function:[eq3]Obviously, the joint moment generating function exists and it is well-defined because the above expected value exists for any $tin U{211d} ^{2}$.

Exercise 1.2

Let [eq4] be a $2 imes 1$ random vector with joint moment generating function[eq5]Derive the expected value of X_1.

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The moment generating function of X_1 is:[eq6]The expected value of X_1 is obtained by taking the first derivative of its moment generating function:[eq7]and evaluating it at $t_{1}=0$:[eq8]

Exercise 1.3

Let [eq9] be a $2 imes 1$ random vector with joint moment generating function[eq10]Derive the covariance between X_1 and X_2.

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We can use the following covariance formula:[eq11]The moment generating function of X_1 is:[eq12]The expected value of X_1 is obtained by taking the first derivative of its moment generating function:[eq13]and evaluating it at $t_{1}=0$:[eq14]The moment generating function of X_2 is:[eq15]To compute the expected value of X_2 we take the first derivative of its moment generating function:[eq16]and evaluating it at $t_{2}=0$:[eq17]The second cross-moment of X is computed by taking the second cross-partial derivative of the joint moment generating function:[eq18]and evaluating it at [eq19]:[eq20]Therefore:[eq21]

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