Question

The incomes of all families in a particular suburb can be represented by a continuous random variable. It is known that the median income for all families in this suburb is $60,000 and that 40% of all families in the suburb have incomes above $72,000.
a) For a randomly chosen family what is the probability that its income will be between $60000 and $72000.

b) If the distribution function for the income is known to be uniform what is the probability that a random chosen family has an income below $65000.

Answer 1

Answer:

a) 0.10 or 10%

b) 0.5417 or 54.17%

Explanation:

a) The median income of $60,000 is at the 50th percentile of the distribution. If 40% if incomes are above $72,000, then an income of $72,000 is at the 60th percentile of the distribution. Therefore, the probability that a family's income will be between $60,000 and $72,000 is:

$P( \ 60,000 \leq X \leq \ 72,000)=0.6-0.5\\P( \ 60,000 \leq X \leq \ 72,000)=0.1 = 10\%$

b) If the distribution is known to be uniform, the probability that a random chosen family has an income below $65,000 is:

$P( X \leq \ 65,000)=0.5+\frac{65,000-60,000}{72,000-60,000}*0.1\\ P( X \leq \ 65,000)=0.5417 =54.17\%$