91Ó°ÊÓ

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{1}{5}, \quad[3,8] $$

Find the future value \(P\) of each amount \(P_{0}\) invested for time period t at interest rate \(k\), compounded continuously. $$ P_{0}=\$ 55,000, \quad t=8 \mathrm{yr}, \quad k=4 \% $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} 4 e^{-4 x} d x $$

Verify Property 2 of the definition of a probability density function over the given interval. $$ f(x)=4 e^{-4 x}, \quad[0, \infty) $$

Find the accumulated future value of each continuous income stream at rate \(\mathrm{R}(t),\) for the given time \(\mathrm{T}\) and interest rate \(k\) compounded continuously. $$ R(t)=\$ 50,000, \quad T=22 \mathrm{yr}, \quad k=2.75 \% $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} e^{x} d x $$

Find \(k\) such that each function is a probability density function over the given interval. Then write the probability density function. $$ f(x)=k x^{2}, \quad[-1,1] $$

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} x e^{x} d x $$

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(y\) -axis. $$ y=x^{2}, x=0, x=3 $$

A dart is thrown at a number line in such a way that it always lands in [0,10] . Let \(x\) represent the number the dart hits. Suppose the probability density function for \(x\) is given by \(f(x)=\frac{1}{50} x, \quad\) for \(0 \leq x \leq 10\) Find \(P(2 \leq x \leq 6),\) the probability that the dart lands in [2,6]