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Expectations

Helle Bunzel

Iowa State University

September 18, 2008

Expectations of Random Variables I

DeÖnition

The expected value of a discrete random variable exists and is deÖned by E [x ] = (^) ∑x xf (x ) , if and only if (^) ∑x jx j f (x ) < ∞

Discrete case: This is the weighted average of the values taken by x.

Clearly (^) ∑x xf (x ) < ∞ is needed. Why is (^) ∑x jx j f (x ) < ∞ needed?

Consider

L =

∞

n= 1

( 1 )n+^1 n

Expectations of Random Variables III

Then, adding (1) and (2), we get

3 2

L = ( 1 + 0 ) +

L = 1 +

Same terms as in (1) but converges to di§erent value. Conclusion: Absolute convergence is required for uniqueness of the expected value.

Expectations of Random Variables I

Example

A life insurance company o§ers a 50-year-old male $1000 face value, one year term life insurance for a premium of $14. Standard mortality tables indicate that the probability that a male in this age group will die within a year is 0.006. What is the expected gain of the insurance company? DeÖne a random variable X. What is the relevant sample space? What are the corresponding probabilities?

E [X ] = 0. 006  ( 986 ) + 0. 994  14 = 8

Note that 8 is not in the sample space. Throwing a dice: What is the expected value?

Expectations of Random Variables II

a (^) X 10 X (^4) b

X^0

3 0 X 20 x 1 x 2 x 3

But as we allow more and more values into the weighted sum, we get

lim n!∞, mesh! 0

n

i = 1

X (^) i^0 f

X (^) i^0

∆xi =

Z

x

xf (^) (x (^) ) dx

Expectations of Random Variables I

Example

A manufacturer mail out surveys each quarter. The proportion of surveys returned each quarter is the outcome of a r.v. with density f (x ) = 3 x^2 I[ 0 , 1 ] (x ) What is the expected proportion of returned survey any given quarter?

E (x ) =

Z

x

xf (x ) dx =

Z (^) ∞

∞

x 3 x^2 I[ 0 , 1 ] (x ) dx

Z (^1)

0

3 x^3 dx =

x^4

0

Expectation of a function of a r.v. I

Examples

Continuous: Consider the following proÖt function: Π (X ) = 5 X 2 X X is a perishable commodity delivered for processing. f (^) (X (^) ) = 1110 +^2 x 1 [ 0 , 10 ] (x (^) ) Expected value of proÖt:

E (Π) =

Z (^) ∞ ∞

( 5 x 2 x ) 1 + 2 x 110 (^1) [ 0 , 10 ] (x ) dx = 3

Z (^10) 0

x 1 + 2 x 110 dx

= 3 110

Z (^10) 0

 x + 2 x^2

 dx = 3 110

 1 2 x^2 + 2 3 x^3

 10

0

3 110

 50 + 2000 3



2150 110 = 13. 77

Expectation of a function of a r.v. II

Examples

Discrete Values 4 6 8 10 12 Mean Y 0. 2 0. 2 0. 2 0. 2 0. 2 8 Consider 1/Y

Values 1 / 4 1 / 6 1 / 8 1 / 10 1 / 12 Mean 1 /Y 0. 2 0. 2 0. 2 0. 2 0. 2 0. 6583 ¯ 6 = ( 1 / 8 )

Note!!!!! E [g (x )] 6 = g (E (x ))

Jensenís inequality I

Theorem

Let X be a r.v. and let g be a continuos function. Then a) E (^) [g (^) (X (^) )]  g (^) (E (^) (X (^) )) if g is convex and E (^) [g (^) (X (^) )] > g (^) (E (^) (X (^) )) if g is strictly convex and X is not degenerate b) E (^) [g (^) (X (^) )]  g (^) (E (^) (X (^) )) if g is concave and E (^) [g (^) (X (^) )] < g (^) (E (^) (X (^) )) if g is strictly concave and X is not degenerate.

Proof of a) Since g is convex, the tangents to g lie below the function.

Jensenís inequality II

Let l (x ) = a + bx be the equation for the tangent to the point (E (X ) , g (E (X ))). Then g (x )  l (x ) = a + bx 8 x and g (E (X )) = a + bE (X ).

Expectations of multivariate r.v.s I

DeÖnition

Let X = (X 1 , X 2 , ..., Xn ) be a multivariate random variable. Then

E [X ] = (E [X 1 ] , E [X 2 ] , ..., E [Xn ]) =

Z ∞

∞

xfX 1 (x ) , ...,

Z (^) ∞

∞

xfXn (x )

Theorem

Let X = (X 1 , X 2 , ..., Xn ) be a multivariate random variable with density function f (x 1 , x 2 , ..., xn ). Then the expectation of Y = g (x 1 , x 2 , ..., xn ) is

E [Y ] = ∑


∑ g (x 1 , x 2 , ..., xn ) f (x 1 , x 2 , ..., xn ) if X is discrete

E [Y ] =

Z (^) ∞

∞

Z (^) ∞

∞

g (x 1 , x 2 , ..., xn ) f (x 1 , x 2 , ..., xn ) dx 1 dx 2 ...dxn if X is continuous

Properties of expectations I

(^1) Let X be a r.v. with density f (^) (x (^) ) and A an event for X. Then E ( (^1) A (x )) = PX (A) (^2) If c is a constant the E (c) = c. (^3) If c is a constant the E (^) (cX (^) ) = cE (^) (X (^) ). (^4) E

h ∑ki = 1 gi (X 1 , X 2 , ..., Xn )

i = ∑ki = 1 E [gi (X 1 , X 2 , ..., Xn )] (^5) If (X 1 , X 2 , ..., Xn ) are independent random variables then E (^) [∏ni = 1 Xi ] = (^) ∏ni = 1 E (^) [Xi ]

Properties of expectations II

Proof of 5.

Z (^) ∞

∞

Z (^) ∞

∞

n

i = 2

xi

! Z

∞ ∞

x 1 fX 1 (x 1 ) dx 1

n

i = 2

fXi (xi )

dx 2 ...dxn

= E (X 1 )

Z (^) ∞

∞

Z (^) ∞

∞

n

i = 2

xi

n

i = 2

fXi (xi )

dx 2 ...dxn

= ...

n

i = 1

E (Xi )

Moments of a random variable I

Two types of moments: Moments about the origin Moments about the mean. (Central moments)

DeÖnition

Let X be a r.v. with density f (x ). Then the r 0 th moment about the origin, denoted by μ^0 r is deÖned as

μ^0 r = E (X r^ )

Note that: μ^00 = 1 μ^01 = E (X )