Question

In this exercise, you are to consider two variables, $x$and $y$defined on a hypothetical population. Following are the conditional distributions of the variable $y$corresponding to each value of the variable$x$.
a. Are the variables $x$and$y$ associated? Explain your answer.
b. Determine the marginal distribution of$y$.
c. Can you determine the marginal distribution of $x$? Explain your answer.

Short answer

(a) No, they are not associated the conditional distribution for y given x are same for all values of x.

(b) The marginal distribution of yis :

$$\begin{gathered} 1-0.422,2-0.211,3-0.047,4-0.004 \\ Total=1 \\ {N}_A=1000 \end{gathered}$$

(c) No, we cannot determine the mrginal distribution of x from the given data.

Step-by-step solution

Part (a) Step 1. Given Information.

the variables x, ydefined on hypothetical population.

Part (a) Step 2. Find if the variables are associated.

No, they are not associated the conditional distribution fory givenx are same for all values ofx.

Part (b) Step 1. Marginal distribution of y.

Marginal distribution of y is same as the conditional, distribution of y for any given value of x.

The marginal distribution of y is:

$$\begin{gathered} 1-0.422,2-0.211,3-0.047,4-0.004 \\ Total=1 \\ {N}_A=1000 \end{gathered}$$

Part (c) Step 1. Marginal distribution of x.

No, we cannot determine the mrginal distribution of x from the given data.

There are two distribution of xthat are compatible with the given data.

1. $N_B=1000,N_C=1000,N_Z$ where x = zis the number of samples with $Z\in \{A,B,C\}where{N}_A=1000$.

2. $N_B=1000,N_C=1000,N_C=1000$

These two cases are compatible with the given data but the marginals of x are different.