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Problem

Let $X_i$ be i.i.d. $Uniform(0,1)$. We define the sample mean as

\begin{align}%\label{} M_n=\frac{X_1+X_2+...+X_n}{n}. \end{align}

1. Find $E[M_n]$ and $\mathrm{Var}(M_n)$ as a function of $n$.
2. Using Chebyshev's inequality, find an upper bound on \begin{align}%\label{} P\left(\left|M_n-\frac{1}{2}\right| \geq \frac{1}{100}\right). \end{align}
3. Using your bound, show that \begin{align}%\label{} \lim_{n \rightarrow \infty} P\left(\left|M_n-\frac{1}{2}\right| \geq \frac{1}{100}\right)=0. \end{align}

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Problem

The number of accidents in a certain city is modeled by a Poisson random variable with an average rate of $10$ accidents per day. Suppose that the number of accidents on different days are independent. Use the central limit theorem to find the probability that there will be more than $3800$ accidents in a certain year. Assume that there are $365$ days in a year.

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Problem

In a communication system, each codeword consists of $1000$ bits. Due to the noise, each bit may be received in error with probability $0.1$. It is assumed bit errors occur independently. Since error correcting codes are used in this system, each codeword can be decoded reliably if there are less than or equal to $125$ errors in the received codeword, otherwise the decoding fails. Using the CLT, find the probability of decoding failure.

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Problem

$50$ students live in a dormitory. The parking lot has the capacity for $30$ cars. Each student has a car with probability $\frac{1}{2}$ , independently from other students. Use the CLT (with continuity correction) to find the probability that there won't be enough parking spaces for all the cars.

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Problem

The amount of time needed for a certain machine to process a job is a random variable with mean $EX_i=10$ minutes and $\mathrm{Var}(X_i)=2$ minutes$^{2}$. The times needed for different jobs are independent from each other. Find the probability that the machine processes less than or equal to $40$ jobs in $7$ hours.

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Problem

You have a fair coin. You toss the coin $n$ times. Let $X$ be the portion of times that you observe heads. How large $n$ has to be so that you are $95\%$ sure that $0.45 \leq X \leq 0.55$? In other words, how large $n$ has to be so that

\begin{equation} P\big(0.45 \leq X \leq 0.55\big) \geq .95 \qquad ? \end{equation}

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Problem

An engineer is measuring a quantity $q$. It is assumed that there is a random error in each measurement, so the engineer will take $n$ measurements and reports the average of the measurements as the estimated value of $q$. Specifically, if $Y_i$ is the value that is obtained in the $i$'th measurement, we assume that

\begin{align}%\label{} Y_i=q+X_i, \end{align} where $X_i$ is the error in the $i$th measurement. We assume that $X_i$'s are i.i.d. with $EX_i=0$ and $\mathrm{Var}(X_i)=4$ units. The engineer reports the average of measurements \begin{align}%\label{} M_n=\frac{Y_1+Y_2+...+Y_n}{n}. \end{align} How many measurements does the engineer need to make until he is $95\%$ sure that the final error is less than $0.1$ units? In other words, what should the value of $n$ be such that \begin{equation} P\big(q-0.1 \leq M_n \leq q+0.1\big) \geq 0.95 \qquad ? \end{equation}

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Problem

Let $X_2$, $X_3$, $X_4$, $\cdots$ be a sequence of random variables such that

\begin{equation} \nonumber F_{X_n}(x) = \left\{ \begin{array}{l l} \frac{e^{n(x-1)}}{1+e^{n(x-1)}} & \quad x>0\\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation} Show that $X_n$ converges in distribution to $X=1$.

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Problem

Let $X_2$, $X_3$, $X_4$, $\cdots$ be a sequence of random variables such that

\begin{equation} \nonumber F_{X_n}(x) = \left\{ \begin{array}{l l} \frac{e^{nx}+xe^n}{e^{nx}+ \left(\frac{n+1}{n}\right) e^n} & \quad 0 \leq x \leq 1 \\ & \quad \\ \frac{e^{nx}+e^n}{e^{nx}+ \left(\frac{n+1}{n}\right) e^n} & \quad x > 1 \end{array} \right. \end{equation} Show that $X_n$ converges in distribution to $Uniform(0,1)$.

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Problem

Consider a sequence $\{X_n, n=1,2,3, \cdots \}$ such that

\begin{equation} \nonumber X_n = \left\{ \begin{array}{l l} n & \quad \textrm{with probability } \frac{1}{n^2} \\ & \quad \\ 0 & \quad \textrm{with probability } 1-\frac{1}{n^2} \end{array} \right. \end{equation} Show that

1. $X_n \ \xrightarrow{p}\ 0$.
2. $ X_n \ \xrightarrow{L^r}\ 0$, for $r<2$.
3. $X_n$ does not converge to $0$ in the $r$th mean for any $r \geq 2$.
4. $ X_n \ \xrightarrow{a.s.}\ 0$.

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Problem

We perform the following random experiment. We put $n \geq 10$ blue balls and $n$ red balls in a bag. We pick $10$ balls at random (without replacement) from the bag. Let $X_n$ be the number of blue balls. We perform this experiment for $n=10, 11, 12, \cdots$. Prove that $X_n \ \xrightarrow{d}\ Binomial\left(10,\frac{1}{2}\right)$.

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Problem

Find two sequences of random variables $\{X_n, n=1,2,\cdots \}$ and $\{Y_n, n=1,2,\cdots \}$ such that

\begin{align}%\label{} &X_n \ \xrightarrow{d}\ X, \\ & \qquad \textrm{and}\\ &Y_n \ \xrightarrow{d}\ Y, \end{align} but $X_n+Y_n$ does not converge in distribution to $X+Y$.

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Problem

Let $X_1$, $X_2$, $X_3$, $\cdots$ be a sequence of continuous random variable such that

\begin{equation} \nonumber f_{X_n}(x) = \frac{n}{2} e^{{\large -n|x|}}. \end{equation} Show that $X_n$ converges in probability to $0$.

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Problem

Let $X_1$, $X_2$, $X_3$, $\cdots$ be a sequence of continuous random variable such that

\begin{equation} \nonumber f_{X_n}(x) = \left\{ \begin{array}{l l} \frac{1}{nx^2} & \quad x>\frac{1}{n}\\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation} Show that $X_n$ converges in probability to $0$.

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Problem

Let $Y_1$, $Y_2$, $Y_3$, $\cdots$ be a sequence of i.i.d. random variables with mean $EY_i=\mu$ and finite variance $\mathrm{Var}(Y_i)=\sigma^2$. Define the sequence $\{X_n, n=2,3,...\}$ as

\begin{equation} X_n=\frac{Y_1 Y_2+ Y_2 Y_3+ \cdots Y_{n-1}Y_n+ Y_n Y_1}{n}, \qquad \textrm{ for }n=2,3,\cdots. \end{equation} Show that $ X_n \ \xrightarrow{p}\ \mu^2$.

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Problem

Let $Y_1$, $Y_2$, $Y_3$, $\cdots$ be a sequence of positive i.i.d. random variables with $0 < E[\ln Y_i]=\gamma<\infty$. Define the sequence $\{X_n, n=1,2,3,...\}$ as

\begin{equation} X_n=(Y_1 Y_2 Y_3 \cdots Y_{n-1}Y_n)^{\frac{1}{n}}, \qquad \textrm{ for }n=1,2,3,\cdots. \end{equation} Show that $ X_n \ \xrightarrow{p}\ e^{\gamma}$.

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Problem Let $X_1$, $X_2$, $X_3$, $\cdots$ be a sequence of random variable such that \begin{align}%\label{eq:union-bound} X_n \sim Poisson(n\lambda), \qquad \textrm{ for }n=1,2,3,\cdots, \end{align} where $\lambda>0$ is a constant. Define a new sequence $Y_n$ as \begin{align}%\label{eq:union-bound} Y_n = \frac{1}{n} X_n, \qquad \textrm{ for }n=1,2,3,\cdots. \end{align} Show that $Y_n$ converges in mean square to $\lambda$, i.e., $ Y_n \ \xrightarrow{m.s.}\ \lambda$.

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Problem

Let $\{X_n, n=1,2,\cdots \}$ and $\{Y_n, n=1,2,\cdots \}$ be two sequences of random variables, defined on the sample space $S$. Suppose that we know

\begin{align}%\label{} &X_n \ \xrightarrow{L^r}\ X,\\ &Y_n \ \xrightarrow{L^r}\ Y. \end{align} Prove that $X_n+Y_n \ \xrightarrow{L^r}\ X+Y$. Hint: You may want to use Minkowski's inequality which states that for two random variables $X$ and $Y$ with finite moments, and $1 \leq p < \infty$, we have \begin{align}%\label{} E \bigg[\big{|}X+Y\big{|}^p \bigg] \leq E\big[|X|^p \big]^{{\large \frac{1}{p}}}+E\big[|Y|^p \big]^{{\large \frac{1}{p}}}. \end{align}

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Problem

Let $X_1$, $X_2$, $X_3$, $\cdots$ be a sequence of random variable such that $X_n \sim Rayleigh(\frac{1}{n})$, i.e.,

\begin{equation} \nonumber f_{X_n}(x)= \left\{ \begin{array}{l l} n^2x \exp\left\{-\frac{n^2x^2}{2}\right\} & \quad x>0\\ 0 & \quad \textrm{otherwise} \end{array} \right. \end{equation} Show that $X_n \ \xrightarrow{a.s.}\ 0$.

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Problem

Let $Y_1$,$Y_2$, $\cdots$ be independent random variables, where $Y_n \sim Bernoulli\left({\large \frac{n}{n+1}} \right)$ for $n=1,2,3, \cdots$. We define the sequence $\{X_n, n=2,3,4, \cdots \}$ as

\begin{equation} X_{n+1}=Y_1 Y_2 Y_3 \cdots Y_n, \qquad \textrm{ for }n=1,2,3,\cdots. \end{equation} Show that $ X_n \ \xrightarrow{a.s.}\ 0$.

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