Random Variables: Discrete and Continuous

Two Types of Random Variables

• Discrete variables are countable, such as the number of students in a class (e.g., 2, 24, 34, or 135, but not 232 or 12.23).
• Continuous variables can take any value within a range, like tire pressure (e.g., 0 psi to bursting point) or height (e.g., 4.5 to 7.2 feet).
• Examples of continuous variables include pressure, height, mass, weight, density, volume, temperature, and distance.
• Most real-life observations consist of numerical data representing random variable values.
• A random variable is a function that maps sample space elements to numbers.
• There are two types of random variables: discrete and continuous.
• Discrete random variables can only assume countable values.
• Continuous random variables can take on any of the countless number of values in an interval.

Probability Distribution for a Discrete Random Variable

• A probability distribution for a discrete random variable is constructed to represent it with a graph, table, or formula.
• It specifies the values a random variable can assume and the probability of each value.
• All probability distributions must satisfy two conditions: P(x) ≥ 0 for all X values and ∑P(x) = 1 for all X values.
• Fair coins example, with X as the number of heads observed, the probability distribution is P(x=0) = 1/4, P(x=1) = 1/2, and P(x=2) = 1/4.

Mean and Standard Deviation of Discrete Random Variables

• Formulas help determine the mean, variance, and standard deviation of a discrete random variable.
• The mean (µ) indicates the average value, and the standard deviation (σ) measures the spread of values.
• The population mean is calculated by Σxp(x).
• The expected gain from an insurance policy is calculated considering the probability of the customer living or dying.
• Variance of a discrete random variable is σ² = Σ (x − μ)2P(x).
• Standard deviation is the square root of the variance: σ = √σ².

Sums and Differences of Independent Random Variables

• Probability distributions can be created from data, simulations, or theoretical probability principles.
• When rolling two dice, the probabilities of each possible sum can be displayed in a table.
• Addition and Subtraction Rules exist for random variables, where µX+Y = µX + µY and µX-Y = µX − µY.
• Variances are added for both the sum and difference of independent random variables (σ²X+Y = σ²X + σ²Y) because variation in each variable contributes to overall variation.
• If you add the same value to all the numbers of a data set, the shape and standard deviation of the data set remain the same, but the value is added to the mean (re-centering).
• If you multiply the numbers of a data set by a constant d and then add a constant c, the mean and the standard deviation of the transformed values are expressed as follows: µc+dx = c + dµx, σc+dx = |d|σx

The Binomial Probability Distribution

• A binomial experiment has 'n' independent, identical trials, each with two outcomes: success ('S') or failure ('F').
• 'p' denotes the probability of 'S', and 'q' (1-p) the probability of 'F', which remains constant across trials.
• The binomial random variable 'X' counts successes in 'n' trials.
• A binomial experiment must have only two outcomes, a fixed number of trials, independent trial outcomes, and the same success probability for each trial.
• Binomial probability is calculated by: P(x = k) = (nk) * p^k * (1 – p)^(n-k)
• The expected value (mean) and standard deviation are: E(x) = μx = np, σx = √np(1-p)

The Poisson Probability Distribution

• Useful for describing the number of events during a specific time or in an area/volume.
• Examples include traffic accidents at an intersection or house fire claims per month.
• The probability that an even occurs in a given time, distrance, area, or volume is the same.
• Each event is independent of all other events.

Geometric Probability Distribution

• You can use the Poisson and binomial distributions to decribe discrete random variables.
• Geometric distribution describes until first success occur
• 4 characteristics of a Geometric Probability Distribution
  • Experiment with sequence of independent trials
  • Outcome each trail, success or failure
  • define geometric number of trails until observe first success
  • Probability p(x) is always the same per trial