STATISTICS AND QUANTITATIVE BIOLOGY | IBB Lecture

Introduction

Z distribution is a special normal distribution with mean 0 and deviations 1.
Quantiles is splitting the distribution into different sizes based on probability mass.
SEM (Standard Error of means) is the same as the variance of the sample.
95% of the normal distribution is within 2 $\sigma$ (standard deviations) of the mean.
0.025 quanta:
Median is the point of the 50% quantile.

$\mu$ and $\sigma$ are usually unknown. Thats why we use Greek letters. They are for the gods to know.
$E(x)$ is the expected value, which is our measurement of the mean. For m samples: $E(x) = \sum_{i=1}^{m} f_i X_i = \sum \frac{1}{m} X_i$
The variance is defined as: $S^2 = E \Big(\sum_{i=1}^{m}(X_i - m)^2 \Big) = \sum \frac{1}{n-1} \Big( x_i - m \Big)^2$
In R:
- dnorm : is the Probaility Density Function.
- pnorm : is the Cumulative Density Function.
- qnorm : is the Inverse CDF. Get $X$ from $p(X)$ .
For CLT, we need random independent and identical (iid) variables. But for large $n$ and finite variance for each of the variables, the identical criteria can be dropped.
For small sample sizes, we use the quantiles of the t-distribution instead of the z-distribution.

We consider the exponential distribution: $f(x, \lambda) = \lambda e^{- \lambda x}$
Maximum Likelihood Estimation of the distribution is done to find the correct parameter. [Curve Fitting]

A $\chi^2$ distribution is defined as a sum of square of $k$ Normal Distributions $\N(0, 1)$ . $k$ is also the degrees of freedom, which is the only parameter that the distribution takes as input. $\chi ^2 = \sum_{i}^{k} Z_i^2$
Chi squared distribution
Emperical Cumulative Distribution Graph
Difference between one tail and two tail
Why is proportions a unique data type?
Fisher test can only be used for 2X2 table, whereas $\chi^2$ test can be used for n-column tables.
Fisher -> you get Hyper-geometric distribution
In R the functions are fisher.test and chisq.test.
Fisher Test: Odds ratio