Random Variables

Discrete and Continuous Random Variables

A variable is a quantity whose value changes. If the value is a numerical outcome of a random phenomenon, the variable is called random variable denoted by a capital letter.

Discrete variable

• Variable whose value is obtained by counting

• Has a countable number of possible values

• Representation: Histogram

• Example: number of students present, number of heads when flipping three coins

Continuous variable

• Variable whose value is obtained by measuring

• Takes all values in a given interval of numbers

• Representation: Density Curve

• The probability that X is between an interval of numbers is the area under the density curve between the interval endpoints

• Examples: height of students in class, time it takes to get to school

Expectation and Variance

Expectation

• Describes the average value(mean) of X, written as E(X) or $\mu$
• $E(X) = \Sigma xP(x)$ for discrete random variables

What is the expected value when we roll a fair die?

• There are six possible outcomes: 1, 2, 3, 4, 5, 6.

• Each of these has a probability of 1/6 of occurring. Let X represent the outcome of the experiment.

• Therefore P(1) = 1/6 (the probability that you throw a 1 is 1/6) P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6

  E(X) = 1×P(1) + 2×P(2) + 3×P(3) + 4×P(4) + 5×P(5) + 6×P(6) => 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 => 7/2 => 3.5

• Expectation is 3.5

Variance

• Describes the spread (amount of variability) around the expectation, written as Var(X)

• $Var(X) = E[ (X – \mu )^2 ] = E(X^2) -\mu ^2$

  • $\mu : Expectation/mean$
  • $E(X^2): \Sigma x^2P(x)$
• Note: Standard deviation $\sigma$ is the square root of variance

Joint, Marginal, and Conditional Probabilities

Marginal probability

• The probability of an event occurring

• Formula: $P(A)$
• Example: the probability that a card drawn is red (p(red) = 0.5)

Condititional probability

• The probability of event A occurring, given that event B occurs or is true

• Formula: $P(A|B)=\frac{P(A\cap B)}{P(B)}$

  • Note: $P(A\cap B)$ can be written as $P(AB)$
• Example: Given that you drew a red card (26 cards), what’s the probability that it’s a four (2 cards). (P(four|red))=2/26=1/13

Joint probability

• The probability of event A and event B occurring

• Formula:

  • $P(A\cap B) = P(A|B) P(B)$
  • $P(A\cap B) = P(B|A) P(A)$
• Example: the probability that a card is a four and red =p(four and red) = 2/52=1/26

Links: - http://www.henry.k12.ga.us/ugh/apstat/chapternotes/7supplement.html - https://revisionmaths.com/advanced-level-maths-revision/statistics/expectation-and-variance - http://www.statisticalengineering.com/joint_marginal_conditional.htm - https://sites.nicholas.duke.edu/statsreview/jmc/