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Unformatted text preview: If x ∈ [k/2n , (k + 1)/2n ), then x diﬀers from k/2n by at most 1/2n . So the last integral diﬀers from
(k+1)/2n xf (x)dx
k/2n by at most (1/2n )P(k/2n ≤ X < (k + 1)/2n ) ≤ 1/2n , which goes to 0 as n → ∞. On the other hand,
(k+1)/2n M xf (x)dx =
k/2n xf (x)dx,
0 which is how we deﬁned the expectation of X .
We will not prove the following, but it is an interesting exercise: if Xm is any sequence of discrete
random variables that increase up to X , then limm→∞ E Xm will have the same value E X .
To show linearity, if X and Y are bounded positive random variables, then take Xm discrete increasing
up to X and Ym discrete increasing up to Y . Then Xm + Ym is discrete and increases up to X + Y , so we
E (X + Y ) = lim E (Xm + Ym ) = lim E Xm + lim E Ym = E X + E Y.
m→∞ m→∞ m→∞ If X is not bounded or not necessarily positive, we have a similar deﬁnition; we will not do the
details. This second deﬁnition of expectation is mostly useful for theoretical purposes and much less so for
Similarly to the discrete case, we have
Proposition 6.2. E g (X ) = g (x)f (x)dx. As in the discrete case,
Var X = E [X − E X ]2 .
As an example of these calculations, let us look at the uniform distribution. We say that a random
variable X has a uniform distribution on [a, b] if fX (x) = b−a if a ≤ x ≤ b and 0 otherwise.
To calculate the expectation of X ,
∞ EX = b xfX (x)dx =
a b 1
b−a x dx =
2 This is what one would expect. To calculate the variance, we ﬁrst calculate
∞ E X2 = b x2 fX (x)dx =
a2 + ab + b2
3 We then do some algebra to obtain
Var X = E X 2 − (E X )2 = (b − a)2
12 7. Normal distribution.
A r.v. is a standard normal (written N (0, 1)) if it has density
√ e−x /2 .
2π 15 = a+b
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This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.
- Spring '09