Probability for QC
I am assuming that you are familiar with the basic concepts of set theory/notation and probability theory. so lets start where the rigor of probability begins to glimpse.
Random variables- although we call them variables ,its actually a function/mapping of the sample space to real numbers. this means that events are mapped to either discrete values(giving rise to discrete random variables) or continuous values. mathematically it could be written as:
let Ω be the sample space.
let X be the random variable.
let R be the set of real numbers.
then this mapping would be written as: X: Ω — >R.
after random variables comes PMF(probability mass functions).
A PMF is another function that maps the range of the random variables(used interchangeably with just “random variables”) to a probability value that arises from our knowledge of the chance of occurrence of the individual events of the random variable. this is a sort of encoding of that knowledge into a number that is neither less than zero nor greater than one. the PMF together with the random variable forms what is called a probability mass distribution realized on a 2 dimensional real number system.
next lets focus on expectation of the random variable, this is just the weighted average of the range of the random variable (really just a fancy way of saying the mean of the random variable). written as :
E(X)=Σ x*p(x) , for all x in X, for all p(x) in PMF.
a good eye would see this ,yes it is a linear combination, we could essentially express the random variable and PMF as a vector ,and find the inner product which will be the expectation.
Variance: variance provides a measure of the dispersion of the random variable about its expectation.
mathematically speaking its the expectation of the squared values of the deviation.
Var(X)=E[(X-E(X))²]=Σ([x-E(X)]²)*p(x), for x in X.
take a moment to analyze the above expression.
Law of Large numbers: the mean of the outcome of the trials of an experiment conducted over and over again tend to become equal to the expectation of the random variable for large number of trials, Equal to E(X) at infinity.
Dirac notation : bras and kets → lets have a vector a in ket notation it will be written as |a> ,the complex conjugate transpose of it will be written in bra notation as <a|.
Alright thats all the info for today ,i’ll see you in the next lecture.