(Solved): 1)Prove that the density for the standard normal distribution f(x)=21e2x2 Satisfies: ...

1)Prove that the density for the standard normal distribution $f(x)=\frac{1}{\sqrt{2?}}{e}^{?\frac{x^2}{2}}$ Satisfies: ${?}_{??}^{?}f(x)dx=1$ 2) Find the expected value of the following game: A coin is tossed until heads appears. You receive $\$2$ to play and that amount is doubled each time tails appears. The game ends when the first $\mathrm{H}$ appears. How much would you pay to play this game?