A factory produces items with a $2\%$ defect rate. If $1000$ items are produced, what is the approximate

probability that there are at most $30$ defective items, using the Poisson approximation of the Binomial

distribution?

A$.$ Approximately $0\mathrm{.}90$

B$.$ Approximately $0\mathrm{.}85$

C$.$ Approximately $0\mathrm{.}95$

D$.$ Approximately $0\mathrm{.}80$

Consider a discrete random variable $D$ with a probability mass function $P(D=d)=\frac{1}{4}$ for din$\{1,2,3,4\},$

and a continuous random variable $U$ uniformly distributed over the interval $0,1.$ The probability density

function of $U$ is $f(u)=1$ for $0u1.$ If $D$ and $U$ are independent, what is the correlation coefficient

between $D$ and $U?$

A$.0\mathrm{.}5$

B$.0\mathrm{.}25$

C$.-0\mathrm{.}5$

D$.0$