# Income percentile ranks are numbers from 0 to 100 that indicate where you sit in the income distribution - for example, 75 would mean you earn more income than three quarters of people, but less than the top quarter. Let's assume a child's expected income percentile is a linear function of their parents' income percentile. It turns out the intercept is enough to pin down the slope. Do you see why? If the intercept is 20 - which means the lowest-income parents have children who end up at the 20-th percentile, on average - what is the slope?

Income percentile ranks are numbers from 0 to 100 that indicate where you sit in the income distribution - for example, 75 would mean you earn more income than three quarters of people, but less than the top quarter. Let's assume a child's expected income percentile is a linear function of their parents' income percentile. It turns out the intercept is enough to pin down the slope. Do you see why? If the intercept is 20 - which means the lowest-income parents have children who end up at the 20th percentile, on average - what is the slope?
This problem is from an app called probability puzzles.
My approach: I realised $y=ax+b$ could be the line for child but we don't have any information about the slope. We know that percentile is uniformly distributed between 0 and 100 so expected value is 50 but I don't see how to proceed from here.
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Step 1
I agree with your linear function: $y=ax+b$, where x is the ith parents' income percentile. And y is the child's expected income percentile. Then the crucial information are
If the intercept is 20 - which means the lowest-income parents have children who end up at the 20th percentile, on average.
This give us the equation $y\left(0\right)=20$. A child cannot end up below the 20th percentile: $y\left(0\right)=a\cdot 0+b=b=20$.
Step 2
Then the highest income parents percentile ($x=100$) can have children who end up at the highest income percentile ($y=100$), on average. It cannot be more, since the codomain of the percentile is between 0 and 100. Thus the equation is
$100=a\cdot 100+20$
Finally, calculate the value of a.