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Random Variables

What

A random variable is a variable used in probability and statistics that can take on a set of possible values. Each possible value of the random variable is associated with a probability. For example, if you flip a fair coin four times, the random variable \(X\) could be the number of heads obtained, which can be 0, 1, 2, 3, or 4.

Random variables can be:

  • Discrete: They take on distinct, separate values (e.g., number of heads in coin flips).
    • \(X\) ~ \(Binomial \quad (P_x)\)
  • Continuous: They can take on any value within a range (e.g., height of people, temperature).
    • \(X\) ~ \(Gaussian\quad(\mathcal{N})\)
  • \(P(X)\) is called a distribution
    • define values of \(X\) (i.e. range and domain).
    • Probability \(P\) of each possible value of \(X\).

How

  1. Define the sample space \(\Omega\), which is the set of all possible outcomes (e.g., all sequences of heads and tails for four coin flips).
  2. Define the random variable \(X\) as a function from the sample space to the set of possible values (e.g., number of heads in the sequence).

  3. Assign probabilities to each possible value of \(X\). For the coin flips, this follows a binomial distribution:

    \[ P(X = k) = \binom{4}{k} \left(\frac{1}{2}\right)^k \left(\frac{1}{2}\right)^{n-k} \]

    where \(k = 0, 1, 2, 3, 4\).

  4. Represent this distribution visually, such as with a histogram showing probabilities for each value of \(X\).

  5. For continuous random variables, use a probability density function (PDF), which can be integrated over intervals to find probabilities.

Why

  • Random variables provide a concise way to summarize complex probability spaces. Instead of listing every possible outcome (which can be huge, e.g., flipping 100 coins), you describe the distribution of the variable (like number of heads).
  • Using random variables allows you to compute probabilities easily using known distributions (binomial, normal, etc.) without enumerating all outcomes.
  • Random variables enable hypothesis testing and parameter estimation with data, such as testing if data follows a binomial or normal distribution and estimating distribution parameters from samples.
  • You can visualize and compute with random variables, including expected values, variances, and probabilities within intervals.
  • They support advanced calculations like integrals or sums over probability distributions for continuous or discrete variables.
  • You can build and apply functions on random variables, propagating uncertainty through systems and performing statistical or engineering analyses.

Example

  • Flip a fair coin 4 times:
    • Sample space \(\Omega\) has 16 outcomes (e.g., HHHH, HHHT, ... TTTT).
    • Random variable \(X = \text{number of heads}\), \(x = \text{values can be 0 to 4}\).

  • Probability distribution of \(X\):

    \[ P(X=0) = \frac{1}{16},\quad P(X=1) = \frac{4}{16},\quad P(X=2) = \frac{6}{16},\quad P(X=3) = \frac{4}{16},\quad P(X=4) = \frac{1}{16} \]

  • Plotting these probabilities creates a binomial distribution histogram.

This abstraction helps summarize and compute probabilities efficiently without listing all outcomes.