Properties of expectation

Properties of expectation

Lessons

We can write out mean as an expected value:
μ=E[X]\mu=E[X]

And likewise for variance:
σ2=\sigma^2= Var(X)(X)

n: number of trials
x: number of success in n trials
p: probability of success in each trial

Binomial:
E[X]=npE[X]=np
Var(X)=np(1p)(X)=np(1-p)

Geometric:
E[X]=1pE[X]=\frac{1}{p}
Var(X)=1pp2(X)=\frac{1-p}{p^2}

Properties of Expectation:

\cdot E[X+a]=E[X]+aE[X+a]=E[X]+a
\cdot E[bX]=bE[X]E[bX]=bE[X]
\cdot E[X+Y]=E[X]+E[Y]E[X+Y]=E[X]+E[Y]

Or in full generality:
\cdot E[X1+X2++Xn]=E[X1]+E[X2]++E[Xn]E[X_1+X_2+ \cdots +X_n ]=E[X_1 ]+E[X_2 ]+ \cdots +E[X_n]

Properties of Variance:
\cdot Var[X+a]=[X+a]= Var[X][X]
\cdot Var[bX]=b2[bX]=b^2Var[X][X]
\cdot Var[X+Y]=[X+Y]= Var[X]+[X]+ Var[Y][Y] if X and Y are independent
  • Introduction
    a)
    What is the expected value and variance for random variables?

    b)
    Properties of Expectation and Variance


  • 1.
    Verifying Expectation and Variance
    If a 6 sided die is rolled what is the expected value shown on the die?

  • 2.
    A certain car breaks down every 50 hours of driving time. If the car is driven for a total of 175 hours;
    a)
    What is the expected number of breakdowns?

    b)
    What is the variance of the breakdowns?


  • 3.
    Clara is trying to make the perfect teapot out of pottery. Each time she attempts to make the perfect teapot she will use a lump of clay and she will succeed with a probability of 0.20. Once she makes the perfect teapot she will stop potting.
    a)
    What is the expected number of lumps of clay Clara will use to make this perfect teapot?

    b)
    What is the variance on the number of lumps of clay Clara will use?


  • 4.
    Properties of Expectation and Variance
    If a 6 sided die is rolled what is the expected value shown on the die? What would be the expected value if 10 die were rolled?

  • 5.
    Suppose we have two independent random variable one with parameters E[X]=4E[X]=4 and Var(X)=3(X)=3, and the other with parameters E[Y]=9E[Y]=9 and Var(Y)=6(Y)=6.
    a)
    What is E[X+Y+2] E[X+Y+2]?

    b)
    What is E[3X+2Y5] E[3X+2Y-5] ?

    c)
    What is Var(3X+2) (3X+2)?

    d)
    What is Var(2(X+Y+1)) (2(X+Y+1)) ?