数学代写|MAST31702 Probability Theory

Data is collected on the weather in Marietta. Each month’s weather is summarized as ” $x$ sunny days, $y$ cloudy days, $z$ rainy days”. (Each day is exactly one of sunny, cloudy, or rainy.)
(a) Find the number of possible summaries of the weather in a 30-day month like September.
(b) If you wanted to know the probability that there are 10 sunny days, 10 cloudy days, and 10 rainy days in September, it would not be correct to take $\frac{1}{n}$, where $n$ is the answer to part (a). That’s because we are not sampling uniformly.

Assume that each kind of weather occurs with probability $\frac{1}{3}$, and that weather on different days is independent. Under this assumption, what is the probability that September’s weather is summarized as “10 sunny days, 10 cloudy days, 10 rainy days?”

A fair coin is flipped 5 times. Given that it lands heads 3 times out of 5 , what is the probability that the first of the five results was heads?
(a) Without doing fancy calculations, make a reasonable guess at the answer.
(It is not $\frac{1}{2}$. If you’re unsure, feel free to confirm your guess with me at any point before the homework is due.)
(b) Let $A$ be the event “the first flip was heads” and let $B$ be the event ” 3 out of the 5 flips were heads”.
Use binomial probability to compute $\operatorname{Pr}[A \cap B]$ and $\operatorname{Pr}[B]$. Then, compute $\operatorname{Pr}[A \mid B]$ using the definition of conditional probability, hopefully confirming your answer to part (a).

One of our definitions of weak convergence is that $\mu_n$ goes to $\mu_{\infty}$ if for all continuous $f$ vanishing at $\infty$
$$
\lim {n \rightarrow \infty} \int f d \mu_n=\int f d \mu{\infty}
$$
Note there is no uniformity over $f$. Consider the stronger property that
$$
\lim {n \rightarrow \infty} \sup {f \in A}\left|\int f d \mu_n-\int f d \mu_{\infty}\right|=0
$$
where $A$ is the set of continuous functions vanishing at $\infty$ which are bounded in absolute value by 1 . Give examples of convergence in distribution where this stronger property fails and also give (nontrivial) examples where this stronger property holds.

In elementary probability books, it is often stated, concerning the Central Limit Theorem, that it is a good approximation if $n \geq 30$. The only precise mathematical interpretation of this that I see is the following statement.
For every fixed interval $I$,
$$
\lim {n \rightarrow \infty} \sup {X_1 \in D}\left|\operatorname{Prob}\left(\frac{S_n}{\sqrt{n}} \in I\right)-\operatorname{Prob}(Z \in I)\right|=0
$$
where $D$ is the set of random variables with mean 0 and variance 1 and $Z$ is a standard normal. This corresponds to having the CLT hold uniformly over the distributions used for the summands.
a. Show that such a statement is false.
b. If $D(\epsilon, C)$ is the set of random variables with mean 0 and variance 1 and with
$$
E\left(\left|X_1\right|^{2+\epsilon}\right) \leq C
$$
show, using the Lindeberg-Feller Theorem that the above statement is true if $D$ is replaced by $D(\epsilon, C)$. This would say that the CLT holds uniformly over distributions used for the summands if we restrict to having a uniformly bounded $2+\epsilon$ moment.

(a). Let $X_1, X_2$ be two independent exponential random variables with parameter 1. Show that the pdf of $X_1-X_2$ is continuous without computing it.
(b). Use what you did in Part (a) to obtain the characteristic function of the Cauchy distribution.