Temple University
Department of Economics
Econ 8009
Hwk 2 Key
1. Distorted O-rings and temperature.

a. Mean number of distorted O-rings

b. Variance of temperature

<== population variance

c. Covariance

2. Pepys and the Binomial

Probability of one or more sixes when six dice are rolled = 1 - Pr{no sixes}

Probability of two or more sixes when 12 dice are rolled = 1- Pr{no sixes}-Pr{1 six}

Probability of three or more sixes when 18 dice are rolled = 1-Pr{no sixes}-Pr{1 six}-Pr{2 sixes}

3. Quantile

We need a value of y such that Ff(y)-0.2 = 0

4. Find a and b in a pdf.

The density must integrate to one.

(4.1)
The mean is 2/3.

(4.2)

From 4.1 we have a = 1 - 1/3 b.  Plug this into 4.2 and solve for b.

5. Joint distirbution of X and Y: Conditional expectations.
From the given information we have

a. Conditional means of Y

b. Conditional mean of XY

c. Unconditional Means

d. Independence: 
Y is not independent of X since the joint probabilities are not equal to the products of the marginals.
e. Mean independence
Y is not mean-independent of X since the conditonal mean of y depends on the value taken by X.
f. X and Y uncorrelated?

Yes, they are uncorrelated since their covariance is zero.

6. Incomes

7. Business starts and failures

a. Variances

b. There was some ambiguity in the posing of the question Probability(VarY/Varx > 2) 
If you take this question literally then the answer is the following: If the two variances are equal then the probability that V(Y) > 2Var(X) is

Alternatively you could ask the following:  If the two variances are equal then what is the probability that we observe a ratio of the sample variances at least as large as the one in the problem.  Then the answer is 

Alternatibvely if V(Y) = 2Var(X), what is the probability that we could see a ratio at least as large as that observed

8. Confidence intervals for starts and failures
a. 90% confidence interval for mean of Fails (X)
 I will accept:

The correct method is to use a Student's t

b. 95% confidence interval for variance of Starts (Y)

9. Insurance Claims 

Do not reject the null hypothesis.