784 CHAPTER 38. PROBABILITY

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0.5

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You can see that the probability that X ≤ 3 is very close to 1.

Example 38.7.10 The multivariate normal is as follows. The random variable X hasvalues in Rp and its density function is of the form

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(2π)p/2 det(Σ)1/2 e−12 (x−m)∗Σ−1(x−m)

Here Σ is the covariance matrix. It is a symmetric matrix with positive eigenvalues andm ∈ Rp is the mean. Thus

P(X ∈ A) =∫

A

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(2π)p/2 det(Σ)1/2 e−12 (x−m)∗Σ−1(x−m)dx

You integrate over the set A the density function. Just as in the case of three dimensions, thisis easier said than done. However, if A has a simple form ∏

pk=1(−∞,ak], then P(X ∈ A) =∫ a1

−∞

∫ a2

−∞

· · ·∫ ap

−∞

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(2π)p/2 det(Σ)1/2 e−12 (x−m)∗Σ−1(x−m)dxp · · ·dx1

Assuming there are no mathematical difficulties, the following is the definition of whatis meant by expectation.

Definition 38.7.11 Let X be a discrete random variable such that P(X = j) = f ( j). Thenif g is some function defined on the values of X, E (g(X))≡∑ j g( j) f ( j) assuming the summakes sense. It is called the expected value of g(X) or simply the expectation of g(X). Incase X is a continuously distributed random variable with density f (x) , the expectation ofg(X) is E (g(X))≡

∫g(x) f (x)dx, assuming the integral makes sense.

The two cases considered above are the discrete and continuously distributed cases forrandom variables. However, this does not include all cases. To do this right, one needsthe notion of the Lebesgue integral and measure spaces and one defines exactly what arandom variable is, a measurable function defined on a measure space, instead of referringto it vaguely in terms of the probability “it” has certain values or lies in some set called an“event”. What does always happen is that, assuming everything makes sense,

E (aX +bY ) = aE (X)+bE (Y )

for two random variables X ,Y and scalars a,b. Also, for any random variable X it mayor may not have a valid expectation, denoted as E (X) but in every case, it makes sense tospeak of P(X ∈ E) where E is some interval or more generally something called a Borelset.