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{\bf Nonparametric Statistics Homework I.}\\
Due date: Two weeks from today. Worth $5\%$ of your final grade.
Please write clearly. Computer printouts should be included but do
not dump everything.
\begin{enumerate}
\item We have 10 numbers $X_1 = 0.1, X_2 = 0.2, \cdots, X_9=0.9,
 X_{10}=1.0$. Test if 10 numbers come from a standard normal distribution
 with mean $0.5$ at $5\%$ level.

\item You have a random binary sequence 0110110110110110 (16
numbers), which comes from a Bernoulli distribution. Test
$$H_0: \mu =0.4 \mbox{  vs.  } H_1: \mu > 0.4,$$
where $\mu$ is the population mean. Justify your answer by
computing P-value.

\item Let $X \sim \mbox{ Unif}(0,1)$. Let $Y$ be a random variable whose
probability density function is $f(x) = 2x$ for $0 <x <1$.
Generate 10,000 $Y$'s from $X$. What formula did you use? Plot the
histogram with 100 bins (do not print out 10,000 numbers).

\item Let $X_i \sim N(0,1)$.  Plot the probability density functions of
$X_{(1)}$ and $X_{(10)}$ by computing the empirical distributions
of $X_{(1)}$ and $X_{(10)}$ (Follow the example of Lecture 4).

\item From Problem 4, numerically estimate $\mathbb{E} X_{(1)}$
and ${\bf Var} X_{(1)}$.


\end{enumerate}




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