Means and spreads by integration

Mathematics · Probability · ages 19-20

Name ______________________ Date ____________

X is uniform on [0, 6], so its density is 1/6 there. Find E[X] by integration.
Answer: ______________
X is uniform on [0, 6], so its density is 1/6 there. Find E[X] by integration.
Answer: ______________
Which formula gives the variance of a continuous variable?
- E[X squared] minus the squared mean
- E[X] minus E[X squared]
- The square root of E[X]
A friend claims linearity of expectation works only for independent variables.
Circle one: True False
X is uniform on [0, 6], so E[X] = 3. Find E[2X + 1].
Answer: ______________
X and Y are dependent with finite means. What is E[X + Y]?
- E[X] times E[Y]
- E[X] + E[Y]
- E[X + Y] needs independence first
X is uniform on [0, 6]. Given E[X squared] = 12, find Var(X).
Answer: ______________
X is uniform on [0, 6]. Given E[X squared] = 12, find Var(X).
Answer: ______________
Pat computes E[X squared] = 12 and reports Var(X) = 12. What is wrong?
- Pat integrated the wrong function
- Pat forgot to divide by the range
- Pat skipped subtracting the squared mean
An exponential model with rate 2 has variance 1 over 4. Give it as a decimal.
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

3 · Integrating x times density 1/6 from 0 to 6 gives 36/12 = 3.
3 · Integrating x times 1/6 from 0 to 6 gives 36/12 = 3.
E[X squared] minus the squared mean · Spread is the mean square minus the square of the mean.
False · The mean of a sum is the sum of the means, always.
7 · Linearity gives 2 x E[X] + 1 = 2 x 3 + 1 = 7.
E[X] + E[Y] · Linearity adds means whether the variables are linked or not.
3 · 12 minus the squared mean 9 leaves 3.
3 · E[X squared] = 12 from integration, minus the squared mean 9, leaves variance 3.
Pat skipped subtracting the squared mean · Variance finishes with minus mean squared: 12 minus 9 is 3.
0.25 · 1 over 4 equals 0.25.