uni/year2/semester1/logseq-stuff/pages/Random Variables.md

• #ST2001 - Statistics in Data Science I
• Previous Topic: Probability
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• Random Variables

  • What is a random variable? #card card-last-interval:: 4 card-repeats:: 2 card-ease-factor:: 2.32 card-next-schedule:: 2022-11-21T20:17:50.156Z card-last-reviewed:: 2022-11-17T20:17:50.157Z card-last-score:: 3
    • A random variable is a function that associates a real number with each element in the sample space.
    • The probability distribution of a random variable X gives the probability for each value of X.
    • A random variable takes a numeric value based on the outcome of a random event.
    • Random variables are denoted by a capital letter - X, Y, Z, etc.
    • A particular value of a random variable will be denoted with a lower case letter - x, y, z, etc.
  • What are the two types of random variables? #card card-last-interval:: 28.3 card-repeats:: 4 card-ease-factor:: 2.66 card-next-schedule:: 2022-12-13T03:20:55.056Z card-last-reviewed:: 2022-11-14T20:20:55.056Z card-last-score:: 5
    • There are two types of random variables:
      • Discrete random variables can take one of a finite number of distinct outcomes.
      • Continuous random variables can take any numeric value within a range of values.

• Probability Distributions

  • Discrete Probability Distributions

    • What is the probability distribution of some discrete random variable X? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:14:46.764Z card-last-score:: 1
      • The set of ordered pairs (x, f(x)) is a probability function, probability mass function (pmf), or probability distribution of the discrete random variable X if, for each possible outcome x:
        • 1. f(x) \geq 0,
        • 2. \displaystyle \sum_n f(x) = 1,
        • 3. P(X = x) = f(x).
    • What is the cumulative distribution function of a discrete random variable X? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-18T00:00:00.000Z card-last-reviewed:: 2022-11-17T20:19:37.665Z card-last-score:: 1
      • The cumulative distribution function is the probability that a random variable X with a given probability distribution will be ^^found at a value less than or equal to^^ x.
      • The cumulative distribution function F(x) of a discrete random variable X with probability distribution f(x) is:
        • F(x) = P(X \leq x) = \sum_{t \leq x} f(t), \text{ for } - \infty < x < \infty

  • Continuous Probability Distributions

    • What is the probability distribution function for a continuous random variable? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-18T00:00:00.000Z card-last-reviewed:: 2022-11-17T20:25:33.722Z card-last-score:: 1
      • The function f(x) is a probability distribution function (pdf) for a continuous random variable X, defined over a set of real numbers, if:
        • 1. f(x) \geq 0, \text{ for all } x \in R,
        • 2. \int^{\infty}_{- \infty} f(x) dx = 1,
        • 3. P(a < X < b) = \int^{b}_{a} f(x)dx.
      • Note: P(X = x) = 0, i.e., there is no area exactly at x.

• Expected Value - Location

  • What is expected value for a discrete random variable? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-19T00:00:00.000Z card-last-reviewed:: 2022-11-18T18:33:23.280Z card-last-score:: 1
    • The average, or expected value of a random variable is denoted by E[X] & \mu.
    • It can be found by summing the products of each possible value multiplied by the probability that it occurs:
      • \mu = E[X] = \sum_x xP(X = x)
  • What is the expected value for a continuous random variable? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-18T00:00:00.000Z card-last-reviewed:: 2022-11-17T20:18:12.187Z card-last-score:: 1
    • A useful summary of interest is the average, or expected value of a random variable.
      • The expected value is denoted by E[X] & \mu.
    • The expected value of a continuous random variable can be found by:
    • \mu = E(X) = \int_{-\infty}^{\infty} xf(x)dx

• Variance, Standard Deviation - Spread

  • What is the variance & hence the standard deviation of a discrete random variable? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.36 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:18:19.897Z card-last-score:: 1
    • The variance of a discrete random variable measures the squared deviation from the mean:
      • \sigma^2 = \text{Var}(X) = E[(X - \mu)^2] = \sum_x (x - \mu)^2 P(X =x)
    • Alternatively, variance can be calculated by:
      • \text{Var}(X) = E(X^2) - E^2(X)
      • Where
        • E(X^2) = \sum x^2P(X = x)
    • Or, more usefully, the standard deviation is:
      • \sigma = \text{sd}(X) = \sqrt{\text{Var}(X)}
    • The standard deviation has the advantage of being in the same units as X (& \mu).
  • What is the variance of a continuous random variable? #card card-last-score:: 1 card-repeats:: 1 card-next-schedule:: 2022-11-19T00:00:00.000Z card-last-interval:: -1 card-ease-factor:: 2.5 card-last-reviewed:: 2022-11-18T18:35:05.927Z
    • The variance of a continuous random variable is:
    • \text{var}(X) = \int_{-\infty}^{\infty} (x - \mu)^2 f(x)dx

• Means & Variances

  • Adding or subtracting a constant from data shifts the mean, but does not change the variance or the standard deviation.
    • E[X +c] = E[X] +c, \ \ \text{Var}(X+c) = \text{Var}(X), \ \ \text{sd}(X + c) = sd(X)
    • E[X -c] = E[X] -c,\ \ \text{Var}(X -c) = \text{Var}(X), \ \ \text{sd}(X - c) = \text{sd}(X)
  • Multiplying a random variable by a constant multiplies the mean by that constant, and the variance by the square of that constant.
    • E[aX] = aE[X], \ \ \text{Var}(aX) = a^2 \text{Var}(X), \ \ \text{sd}(aX) = |a|\text{sd}(X)