### Probability Series

### 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 | \(\frac{1-p}{p}\) | \(\frac{1-p}{p^2}\) |