The average net worth of US residents that don\'t own a cell phone is $23k, it i
ID: 3229204 • Letter: T
Question
The average net worth of US residents that don't own a cell phone is $23k, it is $82k for those that own a single cell phone, and it is $142k for those that own two cell phones. Furthermore, 24% of US residents don't own a cell phone, 72% own a single cell phone, and 4% own two cell phones. (a) What is the average net worth of US residents? (b) Consider the random experiment of asking a randomly selected US resident about his net worth and cell phone ownership. Define a pair of random variables, and show how your answer in part (a) can be formulated as an application of the law of iterated expectations.Explanation / Answer
ans 1)a) let the total US population be m.
total US residents who are not possesing a cell phone=0.24m (24% of X)
similarly US resident having 1 and 2 cellphone are 0.72m and 0.04m respectively
total networth of US resident having no cell phone: 0.24m(23)
total networth of US resident having 1 cell phone: 0.72m(82)
total networth of US resident having no cell phone: 0.04m(142)
average net worth of US citizen= total worth/ total citizen= [0.24(23)m + (0.72)m82 + 142(0.04)m]/m=70.24
B) let X and Y be random variable of net worth and number of cellphone ownership of a random citizen respectively.
asa per law of iterated expectations E(X)=E(E(X|Y))
E(X)=E(networth | no cell phone)P(no cell phone) + E(networth | one cell phone)P(one cell phone) +
E(networth | two cell phone)P(two cell phone)
E(X)=E(X|Y=0)P(Y=0)+E(X|Y=1)P(Y=1) + E(X|Y=2)P(Y=2)
=23(0.24) + 82(0.72) + 142(0.04)
= 70.24
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