QBUS2810 Formulas概率论代写


P(Y \mid X)=\frac{P(Y, X)}{P(X)}
$Y, X$ are independent if $P(Y \mid X)=P(Y)$ for all values of $Y, X$
If $Y \sim \operatorname{Poisson}(\lambda)$ then $E(Y)=V(Y)=\lambda$

Contingency table testing:

Expected values under null hypothesis: $e_{i, j}=\frac{R_{i} C_{j}}{N} ; i=1, \ldots, r ; j=1, \ldots, c$
$H_{0}$ : the two category variables are independent;
$H_{1}$ : the two category variables are related, dependent

Pearson test:

For $r=c=2$, all $e_{i, j} \geq 5$ is required. For $r, c>2$, at least $80 \%$ is required: i.e. $r \times c \times 0.8)$ have $e_{i j} \geq 5$
V=\sum_{i=1}^{r} \sum_{j=1}^{c} \frac{\left(o_{i j}-e_{i j}\right)^{2}}{e_{i j}} \approx \chi^{2}
$V \sim \chi_{(r-1)(c-1)}^{2}$ under the null hypothesis.
The Pearson chi-squared test assumptions:
– The counts in each cell are of independent events.
– The counts follow a Poisson distribution.
– For $r=c=2$, all $e_{i, j} \geq 5$ is required. For $r, c>2$, at least $80 \%$ is required: i.e. $r \times c \times 0.8$ cells have $e_{i j} \geq 5$

Fisher’s exact test:

P(a, b, c, d)=\frac{\left(\begin{array}{c}
a+b \\
c+d \\
N \\
\end{array}\right)}=\frac{(a+b) !(c+d) !(a+c) !(b+d) !}{a ! b ! c ! d ! N !}
Fisher’s exact test assumes that the counts in each cell follow a hyper-geometric distribution. Assessing this aspect is outside the scope of QBUS2810.

Median test

$H_{0}$ : the two group medians are equal
– The two groups are independent of each other.
– The data are iid in each group.
– The data are at least on the ordinal scale, i.e. a median is a permissable location measure.
– The assumptions of Pearson’s OR Fisher’s test hold (depending on which method you use)


$H_{0}$ : the two group population means are equal, i.e. $\mu_{1}=\mu_{2}$
t=\frac{\bar{Y}_{1}-\bar{Y}_{2}}{\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}}=\frac{\bar{Y}_{1}-\bar{Y}_{2}}{S E\left(\bar{Y}_{1}-\bar{Y}_{2}\right)}

Two-sample t-test assumptions:

– Each group’s sample is i.i.d.
– The two groups are independent of each other.
– If $n_{i}<30$ : Each group is normally distributed; OR
If both $n_{i} \geq 30$ then $E\left(Y_{i}^{4}\right)<\infty$ in each group, $i=1,2$ so that the CLT holds.
Under these assumptions the t-statistic approximately follows a Student-t distribution, with degrees of freedom given by a complicated formula (which Python will calculate for us).
SLR model:
Y=\beta_{0}+\beta_{1} X+\varepsilon
TSS $=\operatorname{RegSS}+R S S$
where $T S S=\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}, R S S=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}$ and $\operatorname{RegSS}=\sum_{i=1}^{n}\left(\hat{y}_{i}-\bar{y}\right)^{2}$



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MATH 333: Probability and Statistics
Spring 2021 Coordinated Course Syllabus
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Course Description: Descriptive statistics and statistical inference. Topics include discrete and continuous
distributions of random variables, statistical inference for the mean and variance of populations, and graphical
analysis of data.
Number of Credits: 3
Prerequisites: MATH 112 with a grade of C or better or MATH 133 with a grade of C or better.
Course-Section and Instructors
Course-Section Instructor
Math 333-002 Professor S. Mahmood
Math 333-004 Professor P. Natarajan
Math 333-008 Professor D. Schmidt
Math 333-010 Professor K. Horwitz
Math 333-014 Professor K. Horwitz
Math 333-018 Professor W. Guo
Math 333-028 Professor K. Carfora
Math 333-102 Professor K. Carfora
Office Hours for All Math Instructors: Spring 2021 Office Hours and Emails