Let \(X \sim \mathrm{B}(n, p)\). Use the CLT to find \(n\) such that: $P[X>n / 2] \leq 1-\alpha\(. Calculate the value of \)n\( when \)\alpha=0.90\( and \)p=0.45$.

Verify that the normal distribution, geometric distribution, and Poisson distribution have reproductive property, but the uniform distribution and exponential distributions do not.

In a bombing attack, there is \(50 \%\) chance that a bomb can strike the target. Two hits are required to destroy the target completely. How many bombs must be dropped to give a 99 completely destroying the target?

A random variable \(X\) has the following PMF $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline X=x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline P(X=x) & k & 3 k & 5 k & 7 k & 9 k & 11 k & 13 k & 15 k & 17 k \\ \hline \end{array} $$ 1\. Determine the value of \(k\). 2\. Find \(P(X<4), P(X \geq 5), P(0

Consider polling of \(n\) voters and record the fraction \(S_{n}\) of those polled who are in favor of a particular candidate. If \(p\) is the fraction of the entire voter population that supports this candidate, then \(S_{n}=\frac{\bar{X}_{1}+X_{2}+\cdots+X_{n}}{n}\), where \(X_{i}\) are independent Bernoulli distributed random variables with parameter \(p\). How many voters should be sampled so that we wish our estimate \(S_{n}\) to be within \(0.01\) of \(p\) with probability at least \(0.95\) ?

One urn contains three red balls, two white balls, and one blue ball. A second urn contains one red ball, two white balls, and three blue balls: 1\. One ball is selected at random from each urn. Describe the sample space. 2\. If the balls in two urns are mixed in a single urn and then a sample of three is drawn, find the probability that all three colors are represented when sampling is drawn (i) with replacement (ii) without replacement.