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Visualizing Random Variables

Sometimes we dont want the usual representation of random variables, that is when we use functions. Functions change the horizontal axis of the graph. \(f(x) = x- 10\) is a function which shifts the x axis to the left by 10.

Sometimes these functions can either be one to one , which means that each input has a unique output/ no two outputs are the same OR the functions can be many to one , which means outputs for different inputs can be the same/ two or more outputs are same.

See This for what changes occurs on different types of functions.

Many To One Functions

In the case of many to one functions we add the probabilities when the outputs are the same.

Info

If two variables \(X\) and \(Y\) are independent then their functions \(f(X)\) and \(g(Y)\) will also be independent.

Formulas

You are probably better off mugging up these because its not gonna come in future weeks. Also you will be provided with a forumla sheet in the exam with all the formula required for STATS2.

Two uniformly distributed iid random variables

Sum

Given that \(X,Y \sim Uniform \{1,2,3,4.....n\} , W=X+Y\) \(\implies W \in \{2,3,4,5....2n\}\)

\[P(W=w) = \begin{cases} \frac{w-1}{n^2}, & 2 \leq w \leq n+1 \\ \frac{2n - w + 1}{n^2} & n+2 \leq w \leq 2n \end{cases}\]

Maximum

Given that \(X,Y \sim Uniform \{1,2,3,4.....n\} , Z=\max(X,Y)\)

\(\implies Z \in \{1,2,3,....n\}\)

\[P(Z=z) = \frac{2z-1}{n^2}\]

Sum of n independent bernoulli trials

Let \(X_1 , X_2 , X_3 .... X_n\) be the results of \(n\) i.i.d \(Bernoulli(p)\) trials.

The sum of the n random variables \(X_1 , X_2 , X_3 .... X_n\) is \(Binomial(n,p)\)

Sum of 2 random variables taking integer values

Suppose \(X\) and \(Y\) take integer values and let their joint PMF be \(f_{XY}\). Let \(Z = X+Y\)

Let \(z\) be some integer.

\[\begin{align} P(Z=z) &= P(X+Y=z) \\ &= \sum^{\infty}_{x=- \infty} P(X=x , Y=z-x) \\ &= \sum^{\infty}_{x=- \infty}f_{XY}(x , z-x) \\ &= \sum^{\infty}_{x=- \infty}f_{XY}(z-y , y) \\ \end{align}\]

Convolution

If \(X\) and \(Y\) are independent,

\[ f_{X+Y}(z) = \sum^{\infty}_{x = - \infty}f_{X}f_{Y}(z-x) \]

Two Independent Poisson

Sum

\(Z = X+Y\)

\[f_Z(Z) = \frac{e^{-(\lambda_1 + \lambda_2)} \times (\lambda_1 + \lambda_2)^Z}{Z!} \]

Conditional distribution of X|Z

\[P(X=k|Z=n) = \frac{n!}{k!(n-k)!} \times (\frac{\lambda_1}{\lambda_1 + \lambda_2})^k \times (\frac{\lambda_2}{\lambda_1 + \lambda_2})^{n-k}\]

which is also equals to

\[P(X=k|Z=n) = Binomial(n, \frac{\lambda_1}{\lambda_1 + \lambda_2})\]

given that \(X|Z \sim Binomial(n, \frac{\lambda_1}{\lambda_1 + \lambda_2})\)

Max of CDF of 2 independent random variables

Definition (CDF of a random variable)

Cumulative distribution function of a random variable \(X\) is a function \(F_X : \mathbb{R} \to [0,1]\) defined as

\[ F_{X}(x) = P(X \leq x)\]

Suppose \(X\) and \(Y\) are independent and \(Z = \text{max}(X,Y)\).

\[\begin{align} F_Z(z) &= P(\text{max}(X,Y) \leq z) \\ &= P((X \leq z) \text{and} (Y \leq z)) \\ &= P(X \leq z)P(Y \leq z) \\ &= F_X(z)F_Y(z) \end{align}\]