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Ten years ago at a certain insurance company, the size of claims under homeowner insurance policies had an exponential distribution. Furthermore, 25% of claims were less than \$1000. Today, the size of claims still has an exponential distribution but, owing to inflation, every claim made today is twice the size of a similar claim made 10 years ago. Determine the probability that a claim made today is less than \$1000.

I tried my best but got an answer of 0.199 when the answer is actually 0.134

tinlyx
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Y.Knight
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1 Answers1

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So you are given that the size of claims 10 years ago follows some distribution $X \sim \text{Exp}(\lambda)$. You also know that now all claims are twice of a similar claim 10 years ago, so if $Y$ is the size of claims now then $Y = 2X$. Therefore \begin{align*} \Pr(Y < 1000) &= \Pr(X < 500) \\ &= 1 - \Pr(X \geq 500) \\ &= 1 - e^{-500 \lambda} \\ &= 1 - \sqrt{e^{-1000 \lambda} } \\ &= 1 - \sqrt{\Pr(X \geq 1000)} \\ &= 1 - \sqrt{0.75} \\ &\approx 0.134 \end{align*}

bitesizebo
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