Calculating quartiles of a given PDF

percentilestatistics

I am supposed to calculate the quartiles ( 25th, 50th, 75th percentile ) of the PDF –

f(x) = x*( 2 – e^(-2) ) if 0 < x < 1

f(x) = e^(-2x) if x ≥ 1

How should I go about solving this? Should I convert the PDF into a CDF?

Best Answer

Whilst a CDF makes the problem easier, you can just integrate the function from $0$ to some unknown upper bound, where you let the integral be equal to the percentile. For example, if we are integrating a PDF, say $f(x)$, looking to find what the $75^{th}$ percentile is, we can say.

$\int_0^a f(x).dx = 0.75$

Where we now can solve for $a$

Related Solutions

[Math] Is it possible to find the mean and standard deviation when given the normal z-scores for the bottom 25% and top 25%

Using z-score tables for standard normal variable, $$ z_{0.25} = \mu - \sigma \cdot 0.675 \qquad z_{0.75} = \mu + \sigma \cdot 0.675 $$ You now have two equations for two unknowns.

In[213]:= Solve[{Quantile[NormalDistribution[mu, si], 0.25] == 490, 
  Quantile[NormalDistribution[mu, si], 0.75] == 620}, {mu, si}]

Out[213]= {{mu -> 555., si -> 96.3691}}

[Math] Finding the percentile of a value given other percentiles

Your final observation about the $25$th, $50$th, and $75$th percentiles just says the distribution is not symmetric. Do you have a reason to believe that it should be? Using the standard deviation assumes the distribution is Gaussian, which the evidence contradicts. All you really know is that $350.2$ is somewhere between the $75$th and $90$th percentile. If the events between $260.4$ and $736.0$ are equally distributed, you would have $350.2$ at the $\frac {350.2-260.4}{736.0-260.4}\cdot 15+75\approx 78$th percentile. It is probably rather higher than that-if the data is mostly near the middle it will be. It is hard to give a nice justifiable answer.

Best Answer

Related Solutions

[Math] Is it possible to find the mean and standard deviation when given the normal z-scores for the bottom 25% and top 25%

[Math] Finding the percentile of a value given other percentiles

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