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### Probability Series

Basic Probability Concepts

Conditional Probability and Bayes’ Rule

Discrete Random Variables

Continuous Random Variables

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### Discrete Random Variables

Even though it is named variable, discrete random variable is actually a function that maps the sample space to a set of discrete real values.

• Where
• X is the random variable
• S is the sample space
• R is the set of real numbers

### Probability Mass Function

Probability mass function is the probability defined over a given random variable, i.e., it gives the probability that a discrete random variable is exactly equal to some value in the sample space, S.

• For a random variable X, $p(X=c)$ is denoted as $p(c)$ and the mapping of each value in sample space to their respective probabilities is known as pmf.
• For all values c that are not is sample space $p(c) = 0$ because it is pointing to an empty set.

### Commutative Distribution Function

Probability defined over an inequality such as $X \leq c$ gives probabilities of all the events that satisfy the condition from $-\infty$ to c i.e. the probability value estimate for X less than or equal to c. Mathematically,

### Explaination

• Set has 5 boys and 5 girls.
• 3 kids selected at random but gender not known.
• X is the random variable that denotes number of girls selected.
• Event c such that $X(c) = 3$ is given by set $c = \{(GGG)\}$ i.e. all three girls selected are girls.
• The pmf for random variable X will be as follows:
a p(a) CD(a)
0 1/12 1/12
1 5/12 6/12
2 5/12 11/12
3 1/12 1
• Calculations:
• Number of ways of selecting k kids from total of N kids is given by $C_k^N = \frac{N!}{k! (N-k)!}$ implies $C_3^{10} = \frac{10!}{3! (10-3)!} = 120$ for N = 10 and k = 3.
• value 1 : a = 0 means that no girl is selected which implies that k boys are selected out of the total B boys. The number of ways this can be accomplished is given by $C_k^B = \frac{B!}{k! (B-k)!}$ implies $C_3^5 = \frac{5!}{3! (5-3)!} = 10$ for B = 5 and k = 3.
• For commutative distribution $CD(2) = p(0) + p(1) + p(2) = 11/12$.
• Similarly the value of $p(a)$ and $CD(a)$ at other values of $a$ can be calculated.

### Random Variable Metrics

Expected Value

The average or mean value calculated over all the possible outcomes of the random variable.

$E(X) = \sum_n v_i * p(v_i) \tag{3}$

• X is the random variable.
• $v_i$ is the value the random variable takes with probability $p(v_i)$.
• sample space size is n.
• often represented by $\mu$.
• it is a measure of central tendency of the random variable.
• Some other properties of expected value of a random variable:

Variance and Standard Deviation

Variance gives the dispersion of probability mass around the mean value (i.e. E(X), expected value) of the random variable.

$Var(X) = E((X-\mu)^2) \tag{6}$ $\sigma = \sqrt (Var(X)) \tag{7}$

• $\sigma$ is the standard deviation.
• Some other properties of variance of a random variable:

$Var(X) = E(X^2) - (E(X))^2 \label{8} \tag{8}$ $Var(aX + b) = a^2 Var(X) \label{9} \tag{9}$ $Var(X+Y) = Var(X) + Var(Y) \text{ iff X and Y are independent } \tag{10}$

Derivation of equation \eqref{8}

Derivation of equation \eqref{9} using equations \eqref{5}, \eqref{8}

### Some Specific Distributions

Uniform Distribution

• All outcomes are equally possible.
• Eg: Probability of getting a heads or tails for a fair coin.
• Uniform(N) implies N outcomes and each has probability 1/N.

Bernoulli Distribution

• Used to model the random experiment where each trail has exactly 2 possible outcomes.
• One possible outcome is termed as success and other as failure.
• Parameter $p$ is the probability of success in an experiment.
• Random variable $X \in \{0, 1\}$.
• X = 1 has probability $p$ and X = 0 has probability $1-p$ where 0 is denoting failure and 1 denoting success.

Binomial Distribution

• Used to model $n$ independent trails of Bernoulli Distribution.
• If X follows Binomial(n, p) then, X = k implies, the event of k successes in n independent Bernoulli trials.

$P(X=k) = C_k^n * (p^k) * ((1-p)^{n-k})$

• $C_k^n$ is the number of ways of picking k success events in n total trials.
• $(p^k) * ((1-p)^{n-k})$ gives the combined probability of the n Bernoulli trails.

Geometric Distribution

• Also defined over Bernoulli distribution.
• Models the event of k failures before first success.
• $geometric(p)$ is given by

$P(X=k) = (1-p)^k * p$

where $X = k$ is the event where first success occured after k failures.

• If X and Y follows geometric distribution with same probability p, then $X + Y$ is also a geometric distribution.

Expected Value and Variance for Distributions

Distribution E(X) Var(X)
Uniform $\frac{n+1}{2}$ $\frac{n^2-1}{12}$
Bernoulli $p$ $p(1-p)$
Binomial $np$ $np(1-p)$
Geometric $\frac{1-p}{p}$ $\frac{1-p}{p^2}$

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