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## Instructions:

It is true that data analysis has become an essential part of research in many fields, including medicine, pharmacology, and biology. Statistical methods are used to help researchers make sense of large amounts of data and to draw meaningful conclusions from their findings. In addition, data analysis is also important in many other fields, including finance, consultancy, and the public sector.

To perform statistical analysis on real data sets, researchers often use statistical software packages. These packages allow researchers to perform complex statistical analyses quickly and accurately, and to visualize their results in meaningful ways. Some popular statistical software packages include R, SAS, and SPSS.

When performing statistical analysis, it is important to have a strong understanding of the underlying statistical techniques being employed. This includes understanding concepts such as probability, hypothesis testing, and regression analysis. It is also important to understand how to apply these techniques appropriately to real-world data sets.

Interpreting the results of statistical analyses is also a crucial part of the process. Researchers must be able to communicate their findings effectively to others, including colleagues, stakeholders, and the general public. This requires a strong understanding of statistical concepts as well as effective communication skills.

Overall, statistical analysis is a critical tool for researchers and professionals in many fields. By applying statistical techniques appropriately and interpreting their results effectively, researchers can gain valuable insights into their data and make informed decisions based on their findings.

You roll a fair six sided die repeatedly until the sum of all numbers rolled is greater than 6 . Let $X$ be the number of times you roll the die. Let $F$ be the cumulative distribution function for $X$. Compute $F(1), F(2)$, and $F(7)$.

(15) $F(1)$ : Since you never get more than 6 on one roll we have $F(1)=0$.

$$

\begin{aligned}

& F(2)=P(X=1)+P(X=2): \

& P(X=1)=0 \

& P(X=2)=P(\text { total on } 2 \text { dice }=7,8,9,10,11,12)=\frac{21}{36}=\frac{7}{12} .

\end{aligned}

$$

$F(7)$ : The smallest total on 7 rolls is 7 , so $F(7)=1$.

(15) Suppose $X$ is a random variable with cdf

$$

F(x)= \begin{cases}0 & \text { for } x<0 \\ x(2-x) & \text { for } 0 \leq x \leq 1 \\ 1 & \text { for } x>1\end{cases}

$$

(a) Find $E(X)$.

(b) Find $P(X<0.4)$.

(a) $f(x)=F^{\prime}(x)=2-2 x$ on $[0,1]$. Therefore

$$

\begin{aligned}

E(X) & =\int_0^1 x f(x) d x \

& =\int_0^1 2 x-2 x^2 d x \

& =x^2-\left.\frac{2}{3} x^3\right|_0 ^1 \

& =\frac{1}{3}

\end{aligned}

$$

(b) $P(X \leq 0.4)=F(0.4)=0.4(2-0.4)=0.4(1.6)=0.64$.

A political poll is taken to determine the fraction $p$ of the population that would support a referendum requiring all citizens to be fluent in the language of probability and statistics.

(a) Assume $p=0.5$. Use the central limit theorem to estimate the probability that in a poll of 25 people, at least 14 people support the referendum.

Your answer to this problem should be a decimal.

Let $X \sim \operatorname{binomial}(25,0.5)=$ the number supporting the referendum.

We know that

$$

E(X)=12.5, \quad \operatorname{Var}(X)=25 \cdot \frac{1}{4}=\frac{25}{4}, \quad \sigma_X=\frac{5}{2} .

$$

Standardizing and using the CLT we have $Z=\frac{X-12.5}{5 / 2} \approx \mathrm{N}(0,1)$ Therefore,

$$

P(X \geq 14)=P\left(\frac{X-12.5}{5 / 2} \geq \frac{14-12.5}{5 / 2}\right) \approx P(Z \geq 0.6)=\Phi(-0.6)=0.2743 .

$$

where the last probability was looked up in the $Z$-table.