Population Standard Deviation of a Discrete Random Variable一个离散型随机变量的总体标准偏差.ppt

Population Standard Deviation of a Discrete Random Variable一个离散型随机变量的总体标准偏差
(cont.) Standard Deviation of a DiscrX+a) = SD(X)
Example: a = -1
Var(X+a)=Var(X-1)=Var(X)
SD(X+a)=SD(X-1)=SD(X)
Probability
Great

Good

OK

Economic
Scenario
Profit
($ Millions)
5
1
-4
Lousy

10
P(X=x4)
X
x1
x2
x3
x4
P
P(X=x1)
P(X=x2)
P(X=x3)
Probability
Great

Good

OK

Economic
Scenario
Profit
($ Millions)
5+2
1+2
-4+2
Lousy

10+2
P(X=x4)
X+2
x1+2
x2+2
x3+2
x4+2
P
P(X=x1)
P(X=x2)
P(X=x3)
E(x + a) = E(x) + a; SD(x + a)=SD(x); let a = 2
m=
 =
m=
 =
New Expected Value
Long (UNC-CH) way: (compute from “scratch”)
E(x+2)=12(.20)+7(.40)+3(.25)+(-2)(.15)
=
Smart (NCSU) way:
a=2; E(x+2) =E(x) + 2 = + 2 =
New Variance and SD
Long (UNC-CH) way: (compute from “scratch”)
Var(X+2)=(12-)2()+…
+(-2+)2() =
SD(X+2) = √ =
Smart (NCSU) way:
Var(X+2) = Var(X) =
SD(X+2) = SD(X) =
Rules for E(X), Var(X) and SD(X): multiplying by constant b
E(bX)=b E(X)
Var(b X) = b2Var(X)
SD(bX)= |b|SD(X)
Example: b =-1
E(bX)=E(-X)=-E(X)
Var(bX)=Var(-1X)=
=(-1)2Var(X)=Var(X)
SD(bX)=SD(-1X)=
=|-1|SD(X)=SD(X)
Expected Value and SD of Linear Transformation a + bx
Let X=number of repairs a new computer needs each year. Suppose E(X)= and SD(X)=
The service contract for the computer offers unlimited repairs for $100 per year plus a $25 service charge for each repair.
What are the mean and standard deviation of the yearly cost of the service contract?
Cost = $100 + $25X
E(cost) = E($100+$25X)=$100+$25E(X)=$100+$25*=
= $100+$5=$105
SD(cost)=SD($100+$25X)=SD($25X)=$25*SD(X)=$25*=
=$
Addition and Subtraction Rules for Random Variables
E(X+Y) = E(X) + E(Y);
E(X-Y) = E(X) - E(Y)
When X and Y are independent random variables:
Var(X+Y)=Var(X)+Var(Y)
SD(X+Y)=
SD’s do not add:
SD(X+Y)≠ SD(X)+SD(Y)
Var