Porisma

Worked solution

Transform a mean and variance

Calculate the mean and variance of Y=2X+3Y=2X+3, where XX has distribution

xx0123
P(X=x)P(X=x)1/41/41/81/81/21/21/81/8

CJC · Lecture · Example 13

Read the population statistics from the calculator

Use the same probability-weighted 1-Var Stats workflow: values 0,1,2,30,1,2,3 in L1 and weights 0.25,0.125,0.5,0.1250.25,0.125,0.5,0.125 in L2. Both source screens give E(X)=1.5E(X)=1.5 and σx=1\sigma_x=1, so Var⁡(X)=1\operatorname{Var}(X)=1. The other displayed values are Σx=1.5\Sigma x=1.5, Σx2=3.25\Sigma x^2=3.25, total weight n=1n=1, minX=0 and Q1=0.5, with Sx blank.

How did this part go?

Transform the mean

E(Y)=E(2X+3)=2E(X)+3=2(1.5)+3=6E(Y)=E(2X+3)=2E(X)+3=2(1.5)+3=6.

How did this part go?

Transform the variance

Var⁡(Y)=Var⁡(2X+3)=22Var⁡(X)=4(1)=4\operatorname{Var}(Y)=\operatorname{Var}(2X+3)=2^2\operatorname{Var}(X)=4(1)=4. The shift by three changes the centre but not the spread.

How did this part go?

Compare your sketch

Check a second probability-weighted distribution

  1. 1. Enter outcomes and probabilities

    In STAT → EDIT, place the outcomes in L1 and their probabilities in L2.

    Enter outcomes and probabilities
    L1 · outcomeL2 · probability
    00.25
    10.125
    20.5
    30.125

  2. 2. Calculate the weighted statistics

    Open STAT → CALC → 1-Var Stats. Use List=L1 and FreqList=L2, then Calculate.

    Mean x̄ = 1.5; population σx = 1

  3. 3. Interpret the variance

    Square the stored population standard deviation. The variance is 1. An independent check is 3.25 − 1.5² = 1.

The source n=1 is the total probability weight. Use population σx rather than sample Sx; the original source leaves Sx blank. Apply any later linear transformation to this distribution as specified in the question.

View original calculator screens
Expand diagram

Use the scrollbars or arrow keys to inspect details.

Reconstruction of both printed calculator screens: X values 0,1,2,3 and probability weights.25,.125,.5,.125; mean 1.5, sum x1.5, sum x squared 3.25, population standard deviation 1, total weight 1, min 0, Q1.5, sample Sx blank. Student and Teacher screenshots differ only in the empty-list cursor focus.
Reconstruction of both printed calculator screens: X values 0,1,2,3 and probability weights.25,.125,.5,.125; mean 1.5, sum x1.5, sum x squared 3.25, population standard deviation 1, total weight 1, min 0, Q1.5, sample Sx blank. Student and Teacher screenshots differ only in the empty-list cursor focus.
Reconstruction of both printed calculator screens: X values 0,1,2,3 and probability weights.25,.125,.5,.125; mean 1.5, sum x1.5, sum x squared 3.25, population standard deviation 1, total weight 1, min 0, Q1.5, sample Sx blank. Student and Teacher screenshots differ only in the empty-list cursor focus.
Open original screens (new tab)

Common mistakes

Multiply the variance by the square of the scale factor. Do not add 3 to the variance. The displayed sigma 1 is a standard deviation and must be squared to obtain Var(X)=1.

Read the hint