Problem on Normal Distribution My homework problem is as follows: I start with a string of 0's of length $n$. Then, with probability $p$, I change each bit to 1 identically and independently. After that, I count the number of 1's in the string (which, on average, will be $np$). Let the number of 1's be $P_1$. Continue this process 100 times, get integers $P_1, P_2,\ldots, P_{100}$. I want to estimate maximum in the set $\{P_1, P_2, .., P_{100}\}$. 
 A: You already know that each of the $P_i$ ($i=1,2,\ldots,100$) follows a binomial distribution with parameters $n$ and $p$. According to your comments and your question tags, you also know that each of these binomials can be approximated by a normal distribution. You haven't mentioned the parameters for these normals, but they aren't too hard to determine: it's reasonable to think that these normals should have the same expected value and variance as the original binomial distributions. Each $P_i$ is approximately normally distributed with parameters $\mu=np$ and $\sigma^2=np(1-p)$.
Now you've defined a new random variable $T$ as the maximum of $P_1,\ldots,P_{100}$. I'm assuming you want to estimate the expected value of $T$, $\textrm{E}\left[T\right]$. I've always found the most straightforward approach for similar problems to be the following. 


*

*Try to calculate the cumulative density function for $T$ by rewriting $P(T\leq t)$ in terms of the other random variables in the problem ($P_1,\ldots,P_{100}$). Hint: If the maximum of a set of numbers is bounded above by $t$, what can you say about the bounds on the individuals in the set?

*Calculate the probability density function by differentiating the cumulative density function found in the previous step.

*Now you should be able to write down an integral representing the expected value of $T$. Evaluating the integral may be difficult (as I believe it is in this case), but nothing is stopping you from using numerical integration.


Sometimes it helps to work backwards. You want $\textrm{E}\left[T\right]=\textrm{E}\left[\max(P_1,\ldots,P_{100})\right]$. The expected value of a random variable is written as $\textrm{E}\left[T\right]=\int_{-\infty}^{\infty}tf_T(t)dt$. To calculate the integral you need to know the density of $T$, $f_T(t)$. For me, it's always been more simple to calculate the density by differentiating the CDF, but I'm sure there are other simple ways to calculate it too.
