Overview

The professor is Shahriar Mirzadeh

This class extends the concepts covered in Probability with more advanced topics

Example: Flipping an unfair coin, where $P(X=1)=p$ if a head is flipped
Then $P(X=0)=1-p$

I.e. $X$ follows a Bernoulli distribution

Say we do this $n$ times, $x_1,x_2,\dots ,x_n$ are independently, identically distributed variables ($x_1,x_2,\dots ,x_n\stackrel{\text{i.i.d.}}{\sim }\text{Ber}(p)$)
What’s $P(X_1=1,X_2=0,X_3=1)$? Well these are all independent, so we get $p^2(1-p)$
We can also say generally that the probability of a sequence with $X={\Sigma }_i{x}_i$ heads is $p^X(1-p{)}^{n-X}$

Now let’s go backwards. How do we estimate $p$ from an experiment/observation? $p\approx \hat{p}=\frac{X}{n}$ is our most probable estimate. As expected, we can get this by maximizing our function $p^X(1-p{)}^{n-X}$ (differentiate to find local maxima)

The same principle applies for other distributions $f_{\theta }(x)$, whether a pmf or pdf

Continue with Parameter Estimation