M1: Random Variable/ Probability Distribution

Module Overview

• Module Title: Random Variables and Probability Distributions

• Institutions: FEU Alabang, FEU Diliman, FEU Tech

Types of Random Variables

Discrete Random Variables

• Definition: Outcomes are countable; values form a finite or countable set.

• Examples:

  • Number of pencils in a box

  • Number of soldiers in a troop

  • Number of rotten tomatoes in a basket

  • Number of defective flashlights

Continuous Random Variables

• Definition: Takes on values on a continuous scale; often represents measured data.

• Examples:

  • Amount of antibiotics in a vial

  • Lifetime of light bulbs in minutes

  • Length of wire ropes

  • Voltage of radio batteries

Classification Exercise

• Items to classify:

  • The number of defective computers (Discrete)

  • The weight of newborns each year in a hospital (Continuous)

  • The number of siblings in a family in a region (Discrete)

  • The amount of paint utilized in a building project (Continuous)

  • The number of dropouts in a school district for a period of 10 years (Discrete)

Sample Space and Functions

• Definition: The set of all possible outcomes from an experiment; associated with each outcome is a real number

• Example: Rolling two dice yields 36 outcomes.

Illustrative Examples

Tossing Three Coins

• Random Variable Y: Number of tails

• Sample Space Outcomes: TTT, TTH, THT, HTT, HHT, HTH, THH, HHH

• Corresponding Values (Y): 0, 1, 2, 3 tails

Probability Distributions

• Discrete Probability Distribution: Values a random variable can assume with their associated probabilities.

• Example: Number of tails when three coins are tossed. Possible values Y (0, 1, 2, 3) with probabilities 1/8, 3/8, 3/8, 1/8 respectively.

Computing Average and Variance

Mean of a Discrete Random Variable

• Formula: ( \mu = \Sigma (X \cdot P(X)) )

• Example: When rolling a die, the mean is 3.5.

Variance of a Probability Distribution

• Definition: Describes spread or variability in distribution.

• Formula: ( \sigma^2 = \Sigma (X^2 \cdot P(X)) - (\mu^2) )

• Steps to compute variance:

  1. Find the mean.

  2. Multiply each value by its probability.

  3. Sum the results.

  4. Subtract the square of the mean from the total.

Visualization

• Probability Histogram: Graphical representation of a discrete random variable's probability distribution, values on the horizontal axis and probabilities on the vertical axis.

The General Social Survey asked 827 people how many days they would wait to seek medical treatment if they were suffering pain that interfered with their ability to work. The results are presented in the following table.

Compute the standard deviation of the distribution.

USA Today reported that approximately 25% of all state prison inmates released on parole become repeat offenders while on parole. Suppose the parole board is examining five prisoners up for parole. Let x be the number of prisoners out of five parole who become repeat offenders, and their corresponding probabilities.

What is the summation of the product of the square of x and the probabilities of the distribution?