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\lhead{\textbf{Molly Soja}\\
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\rhead{Solutions to HW6}

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\begin{problem}{}
We throw 2n balls uniformly at random into n bins independently. Let X be the number of non-empty bins.\\ \\
What is the probability that a specific bin is non-empty (Hint: there is a standard discrete distribution that you can apply)?\\ \\
In expectation, how many non-empty bins are there (Hint: Define Xi to be 1 if the i’th bin is non-empty and 0 otherwise. Write X in terms of Xi)? The answer should be in terms of n.\\ \\
Note that 1 − (1 − 1/n)cn ≈ 1 − 1/ec for large n. In expectation, what is the percentage of non-empty bins?\\ \\
\textbf{Solution}: \\


\end{problem}










\begin{problem}{}
Write a program to simulate the experiment in problem 1. Try n ∈ {102, 103, 106}. Report the number of non-empty bins in each case. Does the answer seem to concentrate around the expectation? It is enough to include your code in your solution, i.e., you do not need to submit your code separately.\\
\textbf{Solution}: \\
\end{problem}



\bewpage\pagebreak
\begin{problem}{}

If X and Y are any two random variables, then the covariance of X and Y is defined by Cov[X,Y] = E[(X−E[X])(Y −E[Y])]. Note that Cov[X,X] = Var[X].\\

Show that, if X and Y are independent, then Cov [X, Y ] = 0 (the converse is not true however: Cov[X,Y] = 0 does not imply that X and Y are independent). \\

Show that Var[X+Y]=Var[X]+Var[Y]+2Cov[X,Y]. \\
\textbf{Solution}: \\
\end{problem}




\begin{problem}{}
Let X be a continuous random variable with the distribution given by the following pdf\\

\[ f(x) = \begin{cases}
      x^3 - x^2 & 2\leq x\leq a \\
      0 & otherwise
   \end{cases}
\]

Find a so that f(x) is a valid pdf.\\\\
What is the expected value of X?\\
\textbf{Solution}: \\


\end{problem}
\pagebreak\newpage


\begin{problem}{}
The lifespan of endangered Siberian tigers (in years) approximately follows the normal distribution with parameters μ = 17,σ = 2. Let X be the lifespan of a specific Siberian tiger. Note that X is a continuous random variable (for example., X can be 18.2 years).\\

What is the probability that a Siberian tiger lives longer than 20 years but
no more than 25 years? One way to do it is to go on https://www.wolframalpha.com/ and type “NormalDistribution[μ, σ] from a to b”. \\

WhatisE[X]andVar[X]?\\

Give an upper bound on the probability that a Siberian tiger dies before 13 years old or after 21 years old using Chebyshev inequality. Hint: this simply means X ≤ 13 or X ≥ 21 and so what can you say about $\mid X - E(X) \mid$\\
\textbf{Solution}: \\

\end{problem}







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