Exercise 4 In Neverland, annual income (in $), X, is distributed according to a
ID: 2929432 • Letter: E
Question
Exercise 4 In Neverland, annual income (in $), X, is distributed according to a Gamma distribution with = 5 and -10 000. Every year, the IRS audits 1% of the individuals with an income below $50,000. 3% of individuals with incomes between $50,000 and $95,000, and 6% of individuals with an income above $95,000. Suppose that the individuals to be audited are selected at random. (a) What is the distribution of the income groups? That is, what proportion of Neverland's population falls into each of the three income groups? Even more specifically, find P(X $50,000), P($50,000 $95,000). Hint: These probabilities should add to 1 (b) You overhear Mr. Statman complain about being audited. What is the probability that Mr. Statman's income is below 850,000? Between $50,000 and S95, 000? Above S95,000? Hint: Again, these probabilities should add to 1. Essentially, we're finding the (posterior) distribution of Mr. Statman's possible income (group) given that he's being audited.Explanation / Answer
a)
P(X< 50000) =
=GAMMADIST(50000,5,10000,1)
= 0.559507
P(50000<X< 95000)
= 0.959737 - 0.559807 = 0.400231
P(X> 95000) = 0.040263
b) P(Audit) = 0.559507 *0.01 + 0.03 *0.400231 +0.06 * 0.040263
= 0.02001
P(X<50000|Audit) = 0.559507 *0.01 / 0.02001 = 0.279613
P(50K<X<95K|Audt) = 0.03 *0.400231 /0.02001 =0.6000464
P(X>95K|Audit) = 0.06 * 0.040263 /0.02001 = 0.12072863
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