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flyingsquirrel · #2 April 16th, 2008, 12:12 AM

Hello

(a) The probability that the profit is lower than $u$ is $P(X\leq u)=\int_{-\infty}^uf(x)\,\mathrm{d}x$ . If you want to know the probability that the profit is between $u$ and $v$ ( $u<v$ ), you'll have to evaluate the probability that it is lower than $v$ and greater than $u$ : it boils down to calculating $P(X\leq v)-P(X\leq u) = \int_{-\infty}^vf(x)\,\mathrm{d}x-\int_{-\infty}^uf(x)\,\mathrm{d}x = \int_u^vf(x)\,\mathrm{d}x$ because if the profit is lower than $v$ it is necessarily lower than $u$ .

(b) The expected value of the profit is given by $\int_{-\infty}^{\infty}xf(x)\,\mathrm{d}x$ .

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