Question

Consider the observations jointly taken on the binary random variables X and Y given in the “Problem 1” worksheet in the Table. 1. Organize the data in a two-way table by counting the number of observations that fall within each of the following cells: {X = 0, Y = 0}, {X = 0, Y = 1}, {X = 1, Y = 0}, and {X = 1, Y = 1}.

2. Use observed cell counts found in part 1 to estimate the joint probabilities for (X = x, Y = y)

3. Find the marginal probabilities of X = x and Y = y

4. Find P(Y = 1|X = 1) and P(Y = 1|X = 0)

5. Find P(X = 1 ∪ Y = 1)

6. Use the estimated marginal probabilities found in part 3 above to compute E(X), E(Y ), V ar(X), and V ar(Y ). Do these agree (at least approximately) with the sample average and sample variance of X and Y?

7. Use the estimated joint probabilities found in part 2 above to compute Cov(X, Y ). 8. Are X and Y independent? Explain.

Observation	X	Y
1	0	0
2	0	0
3	0	0
4	0	0
5	0	0
6	0	0
7	0	0
8	0	0
9	0	0
10	0	0
11	0	0
12	0	0
13	0	0
14	0	0
15	0	0
16	0	0
17	0	0
18	0	0
19	0	0
20	0	0
21	0	0
22	0	0
23	0	0
24	0	1
25	1	0
26	0	0
27	0	0
28	0	0
29	0	0
30	0	0
31	0	0
32	0	0
33	0	0
34	0	0
35	0	1
36	1	0
37	0	1
38	1	1
39	1	1
40	1	0
41	0	0
42	0	0
43	0	0
44	0	1
45	1	1
46	1	1
47	1	1
48	1	1
49	1	0
50	0	0

Answer 1

1) The Bi-variate Table is given by-

Table:1

Count	X=0	X=1
Y=0	36	4
Y=1	4	6

2) The joint probabilities for P(X=x,Y=y)-

Table:2

Probabilities	X=0	X=1	Total
Y=0	0.72	0.08	0.80
Y=1	0.08	0.12	0.20
Total	0.80	0.20	1

3) The Marginal Probabilities are given by:
Marginal Probability of X-

Marginal Probability of Y-

4)

5)

6) Computation of Expectations and Variances:

Sample Average:

Sample Variance:

From the above calculations we can say that the Expectation and Variance are approximately equal to the Sample Average and Sample Variance.

7)

Correlation Coefficient:

As, the correlation between X and Y is 0.5(approx.) So we can conclude that X and Y are Moderately Correlated.

X and Y are not independent.

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