Random variable
Random variable
Discrete random variable): A random variable X is said to be discrete if there is a finite list of values a1, a2, …, an or an infinite list of values a1, a2, …. such that P (X = aj for some j) = 1. If X is a discrete r.v., then the finite or countably infinite set of values x such that P (X = x) > 0 is called the support of X.
- function mapping from sample space S to real life
- numeric “summary” of an aspect of the experiments(random).
Probability mass function: The probability mass function (PMF) of a discrete r.v. X is the function pX given by pX (x) = P (X = x). Note that this is positive if x is in the support of X, and 0 otherwise.
Suppose that \(X: S \to A ( A \subseteq R )\) is a discrete random variable defined on a sample space S. Then the probability mass function \(f_x: A \to [0, 1] for X\) is defined as:
\[f_x(x) = Pr(X = x) = Pr({s \in S : X(s) = x })\]Total probability of all hypothetical outcomes:
\[\sum_{x \in A} f_x(x) = 1\]Bernoulli distribution: A random variable x is said to have Bernoulli distribution if x has only 2 possible values 0 and 1.
And P(x=1) = p, P(x=0) = 1-p
Indicator random variable: The indicator random variable of an event A is the r.v. which equals 1 if A occurs and 0 otherwise. We will denote the indicator r.v. of A by IA or I(A). Note that IA ~ Bern(p) with p = P (A).
Binomial(n,p): The distribution of # sucesses in n independent Bern(p) trials, is called Bin(n,p), it’s distribution is given by:
\[P(X=k) = \binom{n}{k} \ p^k\ q^{n-k} \\ X \sim Bin(n,p)\]\[X \sim Bin(n,p) \\ Y \sim Bin(m,p) \\ X + Y \sim Bin(m+n, p)\]
Definitions:
(1): X is # successes in n independent Bern(p) trial.
(2): Sum of indicator random variables :
\[X = X_1 + X_2 + ... + X_n,\ X_j = \begin{cases} 1, & \text{if jth trial succeeds} \\ 0, & \text{otherwise} \end{cases} \\ E(X) = P(A)\]i.i.d = Independent and Identically Distributed
(3): PMF - Probability Mass Function
\[P(X=k) = \binom{n}{k} \ p^k\ q^{n-k}\]R.V.s
CDF: Cummulative distribute function
\[X \le x\ \text{is an event} \\ F(x) = P(X \le x), \text{then F is the CDF of X}\]PMF(for discrete r.v.s):
\[P(X = a_j)\ \text{for all j} \\ p_j = P(X = a_j) \\ p_j \ge 0 \\ \sum_j p_j = 1\]hint: Binomial theorem
\[(x + y)^n = \sum_{k=0}^n \binom{n}{k} \ x^{n-k} \ y^k\]check:
\[p(x=k) = \binom{n}{k} \ p^k\ q^{n-k},\ k \in \{ 0, 1, ..., n \} \\ \sum_{k=0}^{n} \binom{n}{k} \ p^k\ q^{n-k}\ = (p+q)^n = 1\]Binomial Distribution
Hypergeometric Distribution : sampling without replacement : X ~ HGeom(w, b, n)
Having #b black marbles and #w white marbles, pick a sample of n marbles. What’s the distribution of white marbles in the sample?
\[P(X=k) = \frac{\binom{w}{k}\binom{b}{n-k}}{\binom{b+w}{n}} ,\ 0 \le k \le w ,\ 0 \le n-k \le b\]\[\binom{m+n}{r} = \sum_{k=0}^r \binom{m}{k} \binom{n}{r-k}\]
Cumulative Distribution Function (CDF) of an r.v. X is the function \(F_X \ \text{given by} \ F_X(x) = P ( X \le x )\) . When there is no risk of ambiguity, we sometimes drop the subscript and just write F (or some other letter) for a CDF.