数学代写|MAST31702 Probability Theory 👍 👍 - 👍代考代写：100%准时可靠您的作业代写专家

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Prerequisites

Prerequisites
Probability theory I and its prerequesites

Learning outcomes
The main point is to study the principal tools in modern Probability, namely, conditional expectations and martingales, together with some examples

Learning materials
Lecture notes; D. Williams: “Probability with martingales”, R. Durrett: “Probability: theory and examples”

Additional info
Target group
Optional course.

Master’s Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Timing
Recommended time/stage of studies for completion: 1. year

Term/teaching period when the course will be offered: varying

Completion methods
Exam, other methods will be described later

MAST31702 Probability Theory HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Suppose that $X_1, X_2, X_3, \ldots$ are independent and all bounded by 1 , and assume that for any sequence $a_n$ of 1 ‘s and -1 ‘s,
$$
\sum_n a_n X_n
$$
converges a.s. Show that
$$
\sum_n X_n^2<\infty \text { a.s. }
$$
Hints: (i). You may (and should) use the fact, stated in class, that for an independent sequence of mean 0 random variables which are uniformly bounded, then finiteness of the sum of the variances is also necessary for a.s. convergence of the corresponding random series.
(ii). Fubini’s Theorem. You may assume, in the application of Fubini’s Theorem, that the relevant sets are measurable without proving that.

问题 2.

Switch the order of quantifiers and assume (the stronger fact) that $P\left(\sum_n a_n X_n\right.$ converges for every sequence $\left.\left(a_n\right)\right)=1$.
Show then that
$$
\sum_n\left|X_n\right|<\infty \text { a.s. }
$$

问题 3.

(c). Let $X_1, X_2, X_3, \ldots$ be independent with $X_n$ being $\pm 1 / n^{3 / 4}$ each with probability $1 / 2$. Show that for every sequence $a_n$ of 1 ‘s and -1 ‘s, $\sum_n a_n X_n$ converges a.s. but that nontheless there a.s. exists a sequence of $a_n$ ‘s for which $\sum_n a_n X_n$ does not converge.

Remark: This sequence of $a_n$ ‘s for which $\sum_n a_n X_n$ does not converge will depend on the particular outcome of $\omega$. This contrasts in an interesting way from the stated behavior in problem 1 where you almost surely obtain convergence for all values of $t$. (The order of the quantifiers is crucial in all of this stuff.)

问题 4.

Six fair 6-sided dice are rolled.
(a) What is the probability that exactly one die lands 6 ?
(b) What is the probability that at most one die lands 6 ?

问题 5.

In the game Bingo, 75 balls numbered 1 through 75 are drawn at random (without replacement) out of some fancy mechanical device. Each player has a Bingo card with 24 of the 75 numbers on it.

If you are playing Bingo, and 4 numbers have been drawn, what is the probability that exactly 2 of the numbers appear on your Bingo card?
(a) Solve this problem using ordered sampling. That is, assume that the outcomes are ordered 4-tuples $\left(n_1, n_2, n_3, n_4\right)$ where $n_i$ is the $i^{\text {th }}$ number drawn. Find the number of outcomes where exactly 2 of the numbers appear on your Bingo card, and divide by the total number of outcomes.
(b) Solve this problem using unordered sampling. That is, assume that the outcomes are sets ${a, b, c, d}$ where $a, b, c, d$ are the four numbers drawn, with no order given. Find the number of outcomes where exactly 2 of the numbers appear on your Bingo card, and divide by the total number of outcomes.