Lecture 19: Properties of Expectation
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1 Lecture 19: Properties of Expectation Dan Sloughter Furman University Mathematics 37 February 11, The unconscious statistician, revisited The following is a generalization of the law of the unconscious statistician. Theorem If X Y are discrete rom variables with joint probability function p, h : R R, Y = h(x, Y ), then E[Y ] = h(x, y)p(x, y). x y Similarly, if X Y are jointly continuous rom variables with joint density function f, h : R R, Y = h(x, Y ), then E[Y ] = h(x, y)f(x, y)dxdy. Example Suppose X Y have joint probability density function {, if < x < y < 1, f(x, y) =, otherwise. Then, for example, E[X] = y xdxdy = 1 y dy = 1 3
2 E[Y ] = y ydxdy = y dy = 3. Note that, using the marginals f X f Y which we found previously, we could have computed E[X] = E[Y ] = xf X (x)dx = yf Y (x)dx = x(1 x)dx = 1 3 = 1 3 y dy = 3. Example 19.. Suppose X Y are independent, each having a uniform distribution on [, 1]. Then, for example, E[XY ] = xydxdy = 1 ydy = 1 4. Note that E[X] = 1 E[Y ] = 1, so we have, in this case, E[XY ] = E[X]E[Y ]. This is in fact true in general for independent rom variables. 19. Expectations of sums Theorem 19.. For any rom variable X real numbers a b, E[aX + b] = ae[x] + b. Proof. We will assume X is continuous with density f (the proof for discrete X is similar). In that case, E[aX+b] = (ax+b)f(x)dx = a xf(x)dx+b f(x)dx = ae[x]+b. Example Suppose X has a stard normal distribution. Then E[X] = 1 xe x dx
3 = 1 ( = 1 = 1 ( =. lim b lim b xe x dx + 1 ( e x ) b ( 1 + e b xe x dx ) b + lim ) b e x ( ) ) + lim e b + 1 b For σ > < µ <, let Y = σx + µ. Then Y is N(µ, σ ), E[Y ] = σe[x] + µ = µ. Theorem If X is a rom variable with moment generating function ϕ X, a b are real numbers, Y = ax +b, ϕ Y is the moment generating function of Y, then ϕ Y (t) = e tb ϕ X (at). Proof. We have ϕ Y (t) = E[e t(ax+b) ] = E[e atx e tb ] = e tb ϕ X (at). Example Suppose X is stard normal. Then ϕ X (t) = 1 = 1 = 1 = e t = e t. 1 e tx e x dx e 1 (x xt) dx e 1 ((x t) t ) dx e 1 (x t) dx Now let Y = σx + µ, where σ > < µ <. Then Y is N(µ, σ ), ϕ Y (t) = e µt ϕ X (σt) = e µt e σ t = e µt+ σ t. 3
4 Note that so ϕ Y (t) = (µ + σ t)e µt+ σ t ϕ Y (t) = ( (µ + σ t) + σ ) e µt+ σ t, E[Y ] = ϕ Y () = µ E[Y ] = ϕ Y () = µ + σ. Theorem For rom variables X Y any real numbers a b, E[aX + by ] = ae[x] + be[y ]. Proof. Suppose X Y are jointly continuous with joint density f. Then E[aX + by ] = = a (ax + by)f(x, y)dxdy = ae[x] + be[y ]. xf(x, y)dxdy + b yf(x, y)dxdy More generally, for rom variables X 1, X,..., X n real numbers a 1, a,..., a n, we have E[a 1 X 1 + a X + + a n X n ] = a 1 E[X 1 ] + a E[X ] + + a n E[x n ]. Example In this example we illustrate another method for finding the expected value of a binomial rom variable. First, suppose X has a Bernoulli distribution with probability of success p. Then E[X] = (1 p) + 1 p = p. Now suppose X 1, X,..., X n are independent Bernoulli rom variables, each with probabilty of success p, let S n = X 1 + X + + X n. Then S n is binomial with parameters n p. Moreover, E[S n ] = E[X 1 ] + E[X ] + + E[X n ] = np. 4
5 Example Suppose n balls are drawn, without replacement, from an urn containing M red balls N black balls. For k = 1,,..., n, let { 1, if the kth ball is red, X k =, otherwise. Then, for any k = 1,,..., n, E[X k ] = P (X k = ) + 1 P (X k = 1) M(N + M 1)(N + M ) (N + M n + 1) = (N + M)(N + M 1)(N + M n + 1) = M N + M. Hence, if S n = X 1 + X + + X n, then E[S n ] = nm N + M. Note that, as in the previous example, X 1, X,..., X n are Bernoulli variables; however, in this case S n is hypergeometric, not binomial Expectations of products Theorem If X Y are independnent rom variables, then E[XY ] = E[X]E[Y ]. Proof. Suppose X Y are jointly continuous with marginal denisties f X f Y, respectively. Then E[XY ] = = = E[X] yf Y (y) = E[X]E[Y ]. xyf x (x)f Y (y)dxdy yf Y (y)dy xf X (x)dxdy 5
6 More generally, if X 1, X,..., X n are independent rom variables, then E[X 1 X X n ] = E[X 1 ]E[X ] E[X n ]. Example If X Y are independent N(µ, σ ) rom variables, then E[XY ] = E[X]E[Y ] = µ. Theorem Suppose X 1, X,..., X n are independent rom variables, with moment generating functions ϕ X1, ϕ X,..., ϕ Xn, Y = X 1 + X + + X n. Then the moment generating function of Y is Proof. We have ϕ Y (t) = ϕ X1 (t)ϕ X (t) ϕ Xn (t). ϕ Y (t) = E[e t(x 1+X + +X n) ] = E[e tx 1 e tx e txn ] = E[e tx 1 ]E[e tx ] E[e txn ] = ϕ X1 (t)ϕ X (t) ϕ Xn (t). Note that, in particular, if X 1, X,..., X n are idependent identically distributed (i.i.d) rom variables, each with moment generating function ϕ, then the moment generating function of S n = X 1 + X + + X n is ϕ Sn (t) = (ϕ(t)) n. Example Suppose X is Bernoulli with probability of success p. The moment generating function of X is ϕ X (t) = E[e tx ] = (1 p) + pe t. If X 1, X,..., X n are i.i.d. Bernoulli rom variables, each with probability of success p, S n = X 1 +X n + +X n, then S n is binomial has moment generating function ϕ Sn (t) = (1 p + pe t ) n. 6
7 19.4 Uniqueness of moment generating functions We will find the following theorem very useful, although its proof is beyond the scope of this course. Theorem Suppose X Y are rom variables with moment generating functions ϕ X ϕ Y, respectively. If ϕ X (t) = ϕ Y (t) for all t in some interval ( t, t ), where t >, then X Y have the same distribution. Example Suppose X Y are independent binomial rom variables, with parameters n p m p, respectively. If ϕ X is the moment generating function of X, ϕ Y is the moment generating function of Y, ϕ X+Y is the moment generating function of X + Y, then ϕ X (t) = (1 p + pe t ) n, ϕ Y (t) = (1 p + pe t ) m, ϕ X+Y (t) = (1 p + pe t ) n+m. It follows that X + Y is binomial with parameters n + m p. Example Suppose X Y are independent Poisson rom variables, with parameters λ µ, respectively. If ϕ X is the moment generating function of X, ϕ Y is the moment generating function of Y, ϕ X+Y is the moment generating function of X + Y, then ϕ X (t) = e λ(et 1), ϕ Y (t) = e µ(et 1), ϕ X+Y (t) = e (λ+µ)(et 1). It follows that X + Y is Poisson with parameter λ + µ. Example Suppose X Y are independent N(µ X, σ X ) N(µ Y, σ Y ), respectively, rom variables. If ϕ X is the moment generating function of X, ϕ Y is the moment generating function of Y, ϕ X+Y is the moment generating function of X + Y, then ϕ X (t) = e µ X+ σ X t, 7
8 ϕ Y (t) = e µ Y + σ Y t, ϕ X+Y (t) = e ( µ σ X +µ Y + +σ Y )t. It follows that X + Y is N(µ X + µ Y, σ X + σ Y ). Example Suppose X Y are independent gamma rom variables, with parameters m λ n λ, respectively. If ϕ X is the moment generating function of X, ϕ Y is the moment generating function of Y, ϕ X+Y is the moment generating function of X + Y, then ( ) m λ ϕ X (t) = λ t for t < λ, for t < λ, ϕ Y (t) = ϕ X+Y (t) = ( λ ) n λ t ( λ ) m+n λ t for t < λ. It follows that X + Y is gamma with parameters m + n λ. 8
P (x). all other X j =x j. If X is a continuous random vector (see p.172), then the marginal distributions of X i are: f(x)dx 1 dx n
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