C3 w1 (#34)

* new changes

docs/index.rst

@@ -23,7 +23,6 @@ Contents
    probability/random_variable
    probability/discrete_distributions
    probability/continuous_distributions
-   probability/continuous_random_variables
    probability/joint_distributions
    probability/covariance_and_correlation
    probability/estimators_and_sampling_distributions

docs/probability/continuous_distributions.md

@@ -207,7 +207,7 @@ samples then the mean of the samples will end up being very close to normality.
 
 :::
 
-### PDF
+### Probability Density Function
 If we plot the normal distribution density function, it’s curve has the following characteristics:
 
 ```{image} https://miro.medium.com/max/1400/0*8ZymFK4ust4AgKyc.png

docs/probability/hypothesis_testing.md

@@ -7,4 +7,82 @@ kernelspec:
 ```{title} What is hypothesis testing?
 ```
 
-# Hypothesis Testing
+# Hypothesis Testing
+
+## What is Hypothesis Testing?
+- Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter.
+- Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is most
+  often used by scientists to test specific predictions, called hypotheses, that arise from theories.
+
+
+### Steps
+
+There are 5 main steps in hypothesis testing:
+
+1. State your research hypothesis as a null hypothesis and alternate hypothesis (Ho) and (Ha or H1).
+2. Collect data in a way designed to test the hypothesis.
+3. Perform an appropriate statistical test.
+4. Decide whether to reject or fail to reject your null hypothesis.
+5. Present the findings in your results and discussion section.
+
+#### State your null and alternate hypothesis
+After developing your initial research hypothesis it is important to restate it as a null ($H_0$) and alternate ($H_1$)
+hypothesis so that you can test it mathematically.
+
+- The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables.
+- The null hypothesis is a prediction of no relationship between the variables you are interested in.
+
+You want to test whether there is a relationship between gender and height. Based on your knowledge of human physiology,
+you formulate a hypothesis that men are, on average, taller than women. To test this hypothesis, you restate it as:
+
+Ho: Men are, on average, not taller than women.
+Ha: Men are, on average, taller than women.
+
+##### Example
+Let $X_1, X_2, \ldots, X_n$ be a random sample from the normal distribution with mean $\mu$ and variance $\sigma^2$
+
+Example of random sample after it is observed:
+
+$$
+2.73,1.14,3.98,2.15,5,85,1.97,2.54,2.03
+$$
+
+Based on what you are seeing, do you believe that the true population mean $\mu$ is
+
+$$
+
+\begin{align}
+\mu<=3 \\
+or \\
+\mu>3 \\
+\text { The sample is } \overline{\mathrm{x}}=2.799
+\end{align}
+
+$$
+
+
+The Sample mean is 
+
+$$
+\bar{x}=\frac{1}{n} \sum_{i=11}^n X_i
+$$
+
+- We're going to tend to think that $\mu<3$ when $\bar{X}$ is "significantly" smaller than 3.
+- We're going to tend to think that $\mu>3$ when $\bar{X}$ is "significantly" larger than 3.
+- We're never going to observe $\bar{X}=3$, but we may be able to be convinced that $\mu=3$ if $\bar{X}$ is not too far away.
+
+Hypotheses:
+
+$\mathrm{H}_0: \mu \leq 3$
+$\mathrm{H}_1: \mu>3 \quad$ alternate
+
+hypothesis
+- The null hypothesis is assumed to be true.
+- The alternate hypothesis is what we are out to show.
+
+Conclusion is either:
+Reject $\mathrm{H}_0 \quad$ OR $\quad$ Fail to Reject $\mathrm{H}_0$
+
+#### Errors in Hypothesis Testing
+
+##### Type I Error

docs/probability/random_variable.md

@@ -474,4 +474,39 @@ The indicator function of a subset A of a set X is a function.
 
 $\text{Indicator function}_{A}(X) = \mathbf{1}_A(x) =\begin{cases} 1, & \text { if } A \cap X \neq \emptyset \\ 0, & \text { otherwise }\end{cases}$
 
-Notation= $\mathbb{1} _{A}(x)$
+Notation= $\mathbb{1} _{A}(x)$
+
+## Random Sample
+
+A collection of random variables is independent and identically distributed if each random variable has the same
+probability distribution as the others and all are mutually independent.
+
+$$
+\large \text{Random Sample} = $X_1, X_2, X_3, ..., X_n
+$$
+
+Suppose that $X_1, X_2, X_3, ..., X_n$ is a random sample from the Normal distribution with parameters
+$\mu$ and $sigma^2$. **Mu and sigma are same for all random variables**
+
+$$
+X_{1},X_{2}, \ldots, X_{n} \stackrel{\mathrm{iid}}{\sim} N(\mu, \sigma^{2})
+$$
+
+Suppose that $X_1, X_2, X_3, ..., X_n$ is a random sample from the gamma distribution with parameters
+$alpha$ and $\beta$.
+
+$$
+X_{1},X_{2}, \ldots, X_{n} \stackrel{\mathrm{iid}}{\sim} \Gamma(\alpha, \beta)
+$$
+
+### Example
+
+A good example is a succession of throws of a fair coin: The coin has no memory, so all the throws are **independent**.
+And every throw is 50:50 (heads:tails), so the coin is and stays fair - the distribution from which every throw is
+drawn, so to speak, is and stays the same: **identically distributed**.
+
+### Independent and identically distributed random variables (IID)
+
+```{centered} Random Sample == IID
+```
+

docs/probability/what_is_probability.md

@@ -13,10 +13,10 @@ kernelspec:
 
 - Probability is the branch of mathematics that deals with the occurrence of a random event.
 - Probability is the measure of the likelihood of an event to happen.
-- Probability means possibility and probability is the study of randomness and uncertainty.
 
-Probability theory is widely used in the area of studies such as statistics, finance, gambling, artificial intelligence,
-machine learning, computer science, game theory, and philosophy.
+probability is the study of randomness and uncertainty. Probability theory is widely used in the area of studies such
+as statistics, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and
+philosophy.
 
 ### Applications of probability
 

docs/requirements.txt

@@ -1,5 +1,6 @@
 torch~=1.12
 seaborn~=0.11
+scipy~=1.9
 myst-nb~=0.16
 sphinx-design~=0.2
 sphinx-copybutton