Econ1150 - PS1
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Econ1150 - PS1
Due: Wednesday, January 25, 10:30AM
1. A few years ago the news magazine The Economist listed some of the stranger explanations used in the past to predict presidential election outcomes. These included whether or not the hemlines of women’s skirts went up or down, stock market performances, baseball World Series wins by an American League team, etc. Thinking about this problem more seriously (i.e. looking for causality rather than correlation), you decide to analyze whether or not the presidential candidate for a certain party did better if his party controlled the house. Accordingly you collect data for 34 presidential elections. You should think of this data as comprising a population which you want to describe, rather than a sample from which you want to infer behavior of a larger population. You generate the accompanying table:
Joint Distribution of Presidential Party Affiliation and Party Control of House of Representatives, 1860-1996
|
Dem. Control of House |
Rep. Control of House |
Total |
|
(Y = 0) |
(Y = 1) |
|
Dem. President (X = 0) |
0.412 |
0.030 |
0.442 |
Rep. President (X = 1) |
0.176 |
0.382 |
0.558 |
Total |
0.588 |
0.412 |
1.00 |
(a) Interpret one of the joint probabilities and one of the marginal probabilities. (b) Compute E (X). How does this differ from E (X | Y = 0)? Explain..
(c) If you picked one of the Republican presidents at random, what is the probability that during his term the Democrats had control of the House?
(d) What would the joint distribution look like under independence? Check your results by calculating the two conditional distributions and compare these to the marginal distribution.
2. Find the following probabilities:
(a) Y is distributed χ4(2) . Find Pr (Y > 9.49) .
(b) Y is distributed to . Find Pr (Y > -0.5) .
(c) Y is distributed F4 ,o . Find Pr (Y < 3.32) .
(d) Y is distributed N (500, 10000). Find Pr(Y > 696 or Y < 304).
3. X and Z are two random variables. X is equal to 1 with probability 0.3, and equal to 0 with probability 0.7. You also know that E [Z | X = 1] = 10, and E [Z | X = 0] = 1. Compute E [Z].
4. This exercise should help you understanding the properties of summations . Remember that
n
Xi = X1 + X2 + ... + Xn _ 1 + Xn
i=1
Consider the following sequences of variables
X1 = 1 X2 = 0 X3 = 2
Y1 = 1
Y2 = 2
Z1 = 3
Z2 = 3
Z3 = 3
Z4 = 3
Z5 = 3
You should show each of the following things two ways, first using the formulas and then using the actual numbers.
(a) Show that
5
Zi = 5Z1
i=1
(b) Show that
3 2 / 3 、 ╱ 2 、 2 / 3 、
(c) Show that
i1 / 、 =
(d) Show that
3 4 3
Xi Zj = 12 Xi
i=1 j=1 i=1
2023-04-29