Please consider the following problem and my answer to it. The answer seems to low to me. Is my answer right?
Thanks,
Bob
Problem:
The median net worth of a certain population is $97K$. To be in the top $5\%$ you need to have a net
worth of $2387K$. Assuming the population follows the Pareto distribution, what is the minimum net
worth you need to be in top $1\%$?
Answer:
For the Pareto distribution we have:
\begin{eqnarray*}
P(X > x) &=& \Big( \frac{x}{x_m} \Big) ^ \alpha \\
\end{eqnarray*}
The above formula assumes that $x$ is greater than $0$. Now, we need to find the two parameters
$x_m$ and $\alpha$.
\begin{eqnarray*}
\Big( \frac{97}{x_m} \Big) ^ \alpha &=& 0.5 \\
\Big( \frac{2387}{x_m} \Big) ^ \alpha &=& 0.95 \\
\end{eqnarray*}
Using software, I find:
\begin{eqnarray*}
x_m &=& 3083 \\
\alpha &=& 0.200386 \\
\end{eqnarray*}
Now we have our model.
\begin{eqnarray*}
0.99 &=& \Big( \frac{x}{x_m} \Big) ^ \alpha \\
0.99 &=& \Big( \frac{x}{3083} \Big) ^ {0.200386}\\
0.9510821 &=& \Big( \frac{x}{3083} \Big) \\
x &=& 3083( 0.9510821 ) \\
x &=& 2932.1861 \\
\end{eqnarray*}
Here is an updated solution:
For the Pareto distribution we have: \begin{eqnarray*} P(X > x) &=& \Big( \frac{x_m}{x} \Big) ^ \alpha \\ \end{eqnarray*} The above formula assumes that $x$ is greater than $0$. Now, we need to find the two parameters $x_m$ and $\alpha$. \begin{eqnarray*} \Big( \frac{x_m}{97} \Big) ^ \alpha &=& 0.5 \\ \Big( \frac{x_m}{2387} \Big) ^ \alpha &=& 0.95 \\ \end{eqnarray*} Using software, I find: \begin{eqnarray*} x_m &=& 3083.33 \\ \alpha &=& -0.200386 \\ \end{eqnarray*} Now we have our model. \begin{eqnarray*} 0.99 &=& \Big( \frac{x_m}{x} \Big) ^ \alpha \\ 0.99 &=& \Big( \frac{3083.33}{x} \Big) ^ { -0.200386 } \\ 1.010101 &=& \Big( \frac{3083.33}{x} \Big) ^ { 0.200386 } \\ 1.00514339 &=& \Big( \frac{3083.33}{x} \Big) \\ x &=& 2932.50 \\ \end{eqnarray*} Is this answer right? My feeling is that it is still too low? If so, what did I do wrong?
Wikipedia says the CDF is $1 - (x_m/x)^\alpha,$ for $x > x_m,$ so you should check your formula for $P(X > x).$ Another reason to check is that your formula gives probabilities that exceed 1/2.
Suggest correcting your formula, re-estimating parameters, and trying again.