(1) A population consists of 3 households, labeled 1,2, and 3. The number of per
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Question
(1) A population consists of 3 households, labeled 1,2, and 3. The number of persons livingin each household and their incomes are given in the following table:
Household
Number of inhabitants
Incomes
1
1
$10,000
2
2
$10,000 , $20,000
3
3
$10,000 , $20,000 , $30,000
A household is selected at random with all households equally likely. Then a person isselected from the household with all people equally likely. Let X denote the numberof the selected household, and let Y denote the income of the selected person.
(i) Write down the probability mass function (frequency function) for X. This is thefunction pX dened by pX(x) = P[X = x] for each possible x.
(ii) Write down the conditional probability mass function (frequency function) for Ygiven X. This is the function pY |X dened by pY |X(y|x) = P[Y = y|X = x] foreach possible x, y pair.
(iii) Write down the joint probability mass function (frequency function) for X, Y. Thisis the function pXY dened by pXY (x, y) = P[X = x, Y = y] for each possible x, ypair.
(iv) Find the probability mass function (frequency function) for Y.
(v) What is the conditional probability mass function (frequency function) for Xgiven Y ? Given that the income of the sampled person is $10,000, what is theprobability distribution of the house number?
(vi) Find the expected value of X and the expected value of Y.
(vii) Find the variance of X and the variance of Y.(viii) Find the covariance between X and Y.
Household
Number of inhabitants
Incomes
1
1
$10,000
2
2
$10,000 , $20,000
3
3
$10,000 , $20,000 , $30,000
Explanation / Answer
Here there are 3 varaibles: housenumber, number of persons, income.You did not mention which variable is to be taken as X and which varaible as Y
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