Question
An organiser's compensation claims have the following distribution: $0 with probability 15%, $100 with 30%, $300 with 40%, and $500 with 15%. The organiser wants the smallest cash reserve \(r\) that fully covers at least 80% of individual claims, where a claim is covered when its amount is less than or equal to \(r\).
What reserve should be kept? A colleague proposes interpolating between payout amounts to obtain a reserve somewhere between $100 and $300. Explain precisely why that calculation is or is not valid for this discrete 80th percentile.
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Hint 1
List the cumulative probabilities at the payout amounts, using the inclusive event \(X\le r\).
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Solution
Let \(X\) denote the compensation claim. The requested lower 80th percentile is defined as
\[q_{0.8}=\inf\{r:\Pr(X\le r)\ge0.8\}.\]
The cumulative probabilities at the possible claim amounts are \(\Pr(X\le0)=0.15\), \(\Pr(X\le100)=0.15+0.30=0.45\), \(\Pr(X\le300)=0.45+0.40=0.85\) and \(\Pr(X\le500)=1\).
Every reserve \(r<300\) can cover at most the $0 and $100 claims, or 45% of claims. Raising the reserve to exactly $300 covers 85%, meeting the 80% rule. Therefore the smallest qualifying reserve is $300. A reserve slightly below $300 would still cover only 45%, so this threshold is genuinely minimal.
Linear interpolation of a cumulative probability between $100 and $300 would implicitly treat part of the 40% probability at exactly $300 as if it were spread continuously over intermediate dollar amounts. It is not: the entire 40% atom becomes covered together when \(r\) reaches $300. Under the explicitly defined \(\inf\{r:F(r)\ge0.8\}\) quantile, interpolation is mathematically inappropriate. Other software quantile conventions can interpolate sample ranks, but they answer a different definition and do not determine the smallest reserve achieving 80% coverage here.
Assumptions
The four listed claim amounts are the complete discrete distribution; their probabilities sum to 100%.
The reserve may be any nonnegative dollar amount, and coverage includes equality \(X=r\).
The target is the smallest threshold with \(\Pr(X\le r)\ge0.8\), not an interpolated sample quantile.
Final Answer
$300. The lower 80th percentile is the first payout threshold whose cumulative probability reaches 80%; interpolation below $300 would cover only 45%.