Statistics-lab™可以为您提供usf.edu MATH5472 Statistical inference统计推断课程的代写代考和辅导服务！

MATH5472 Statistical inference课程简介

That sounds like a challenging and exciting course! Here are some key topics that might be covered in such a course:

1. Probability theory: This would include understanding the axioms of probability and basic rules of probability, such as conditional probability and Bayes’ theorem.
2. Statistical inference: This would cover topics such as estimation, hypothesis testing, and confidence intervals. Students would learn about point estimation, maximum likelihood estimation, and Bayesian estimation. They would also learn about hypothesis testing, including null and alternative hypotheses, p-values, and type I and type II errors.

PREREQUISITES 

1. Linear regression: This would involve understanding the basics of simple and multiple linear regression, including the assumptions and limitations of the models. Students would learn about least squares estimation, hypothesis testing, and confidence intervals.
2. Bayesian statistics: This would cover the principles of Bayesian inference, including Bayes’ theorem, prior and posterior distributions, and Bayesian model selection. Students would learn about Markov Chain Monte Carlo (MCMC) methods for sampling from complex posterior distributions.
3. Machine learning: This would include an introduction to basic machine learning techniques such as decision trees, random forests, and support vector machines. Students would learn about the trade-offs between different types of models, as well as methods for model selection and evaluation.
4. Time series analysis: This would involve understanding the basics of time series models, including autoregressive (AR), moving average (MA), and autoregressive integrated moving average (ARIMA) models. Students would learn about forecasting and time series model selection.
5. Cyber systems modeling: This would cover modeling of complex cyber systems using mathematical tools, such as graph theory and network analysis. Students would learn about the basics of cyber security and how to model and analyze cyber attacks using statistical methods.

Overall, this course would provide a strong foundation in mathematical statistics and its applications to complex cyber systems, data analytics, and Bayesian intelligence.

MATH5472 Statistical inference HELP（EXAM HELP， ONLINE TUTOR）

(a) The derivatives are
$$
\begin{aligned}
G^{\prime}(t) & =\sum_{n=1}^{\infty} n t^{n-1} p_n \
G^{\prime \prime}(t) & =\sum_{n=2}^{\infty} n(n-1) t^{n-2} p_n \
& \vdots \sum_{n=k}^{\infty} \frac{n !}{(n-k) !} t^{n-2} p_n
\end{aligned}
$$ At $t=0$ all terms except the first are zero, so$$

\begin{aligned}

& G(0)=p_0 \

& G^{\prime}(0)=p_1 \

& G^{\prime \prime}(0)=2 p_2 \

& \vdots \

& G^{(k)}(0)=k ! p_k .

\end{aligned}

$$

So $p_k=G^{(k)}(0) / k$ !. This is the reason $G$ is called the probability generating function.

(b) For the binomial distribution

$$

G(t)=\sum_{k=0}^n t^k\left(\begin{array}{l}

n \

k

\end{array}\right) p^k(1-p)^{n-k}=\sum_{k=0}^n\left(\begin{array}{l}

n \

k

\end{array}\right)(t p)^k(1-p)^{n-k}=(t p+1-p)^n

$$

by the binomial theorem.

(c) For the geometric distribution

$$

G(t)=\sum_{n=1}^{\infty} t^n p(1-p)^{n-1}=t p \sum_{n=1}^{\infty}[t(1-p)]^{n-1}=\frac{t p}{1-t(1-p)} .

$$

The $n$-th moment of this density is
$$
\begin{aligned}
E\left[X^n\right] & =\int_1^{\infty} x^n \frac{\alpha}{x^{\alpha+1}} d x=\int_1^{\infty} \alpha x^{n-\alpha-1} d x \
& =\left{\begin{array}{ll}
{\left[\frac{\alpha}{n-\alpha} x^{n-\alpha}\right]_1^{\infty}} & \text { for } \alpha \neq n \
{[\alpha \log (x)]_1^{\infty}} & \text { for } \alpha=n
\end{array}= \begin{cases}\infty & \text { for } \alpha \leq n \
\frac{\alpha}{\alpha-n} & \text { for } \alpha>n .\end{cases} \right.
\end{aligned}
$$

So

$$

E[X]= \begin{cases}\infty & \text { for } \alpha \leq 1 \ \frac{\alpha}{\alpha-1} & \text { for } \alpha>1\end{cases}

$$

and

$$

E\left[X^2\right]= \begin{cases}\infty & \text { for } \alpha \leq 2 \ \frac{\alpha}{\alpha-2} & \text { for } \alpha>2\end{cases}

$$

The variance of $X$ is therefore infinite if $\alpha \leq 2$, and is

$$

\operatorname{Var}(X)=E\left[X^2\right]-E[X]^2=\frac{\alpha}{\alpha-2}-\left(\frac{\alpha}{\alpha-1}\right)^2=\frac{\alpha}{(\alpha-2)(\alpha-1)^2}

$$

for $\alpha>2$.

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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