Paid a dollar or the draw

EasyExpected Value Statistics~3m

You draw $X$ from a continuous Uniform distribution on $[0,1]$ . If $X > \tfrac12$ you are paid a flat $1; otherwise you are paid $X$ dollars. What is the expected payout (in dollars, to four decimals)?

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Split at 1/2: a flat-$1 branch weighted by $P(X>1/2)$ , plus $\int_0^{1/2} x\,dx$ for the paid-X branch.
The integral $\int_0^{1/2} x\,dx = 1/8$ already carries the density; add it to $\tfrac12\times 1$ .

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0.625 (± 0.001)

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Asked at: Multi-Strategy Quant, Mixed Quant & Coding

Paid a dollar or the draw

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