17. a. You have numbers .1, ...,N. The mean is * = ~ Enz, and the variance is i. Show that Of = > Enzian -x2. ii. Show that for any set of numbers, the average of the squares is as large as the square of the average. iii. How can you compute the variance with just one pass through the data *1, . .., ON. b. You have a stream of independent uniform random variables taking on values in [0, 1]. i. How will you generate a uniform random variable in the interval [a, b]? ii. How will you generate a Gaussian random variable with mean 0 and variance 1? iii. How will you generate a Gaussian random vector X with mean vector / and covariance matrix E? 18. The density function of the exponential random variable X with parameter @ > 0 is f (2 ) = if a _ 0 if a 0, for anyo e R. Prove that b. Show that the expected value and the variance of the exponential random variable X are E(X) = _ and Var(X) = c. Show that the cumulative density function of X is F (2) = le- ifx 20 otherwise d. Show that P ( X > t ) = / f ( x)dix = e-at Note: this result is used to show that the exponential variable is memoryless, i.e., P(X > t +s| X > t) = P(X > >)