Probability and Statistics: Exam Review Solutions (MQT-1102)

Probability and Statistics: Exam Review Solutions

Introduction

This study guide summarizes key concepts, definitions, and problem-solving techniques from a college-level Probability and Statistics exam review. Topics include probability calculations, Poisson distribution, measures of central tendency and dispersion, combinatorics, and expected value for discrete random variables.

Probability Theory

Basic Probability Calculations

Probability quantifies the likelihood of an event occurring, expressed as a number between 0 and 1.

• Probability of Event A: $P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total number of possible outcomes}}$

• Example: If there are 4 white balls out of 12, $P(A) = 4/12$.

• Probability of Event B: $P(B) = 3/12$

• Probability of Event C: $P(C) = 8/12$

Incompatible (Mutually Exclusive) Events

• Definition: Two events are incompatible if they cannot occur simultaneously, i.e., $A \cap B = \emptyset$.

• Example: Drawing both a white and a black ball in a single draw is impossible; thus, the events are incompatible.

Independent Events

• Definition: Events A and B are independent if $P(A \cap B) = P(A) \times P(B)$.

• Example: If $P(B \cap C) \neq P(B) \times P(C)$, then B and C are not independent.

Poisson Distribution

Modeling Event Counts

The Poisson distribution models the number of times an event occurs in a fixed interval of time or space.

• Probability Mass Function: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$, where $\lambda$ is the average rate.

• Example: If the average number of items sold in 8 hours is 5.1, then $\lambda = 5.1$.

Calculating Probabilities

• Probability of more than 2 items sold:

  • $P(X > 2) = 1 - P(X = 0) - P(X = 1) - P(X = 2)$

  • Substitute values using the Poisson formula.

• Probability of selling 1 or 2 items:

  • $P(Y = 1) + P(Y = 2)$, where $Y$ is the number of items sold in a given interval.

Descriptive Statistics

Measures of Central Tendency

• Median: The middle value in an ordered data set. If the number of observations is even, the median is the average of the two central values.

  • $\text{Median} = \frac{\text{12th value} + \text{13th value}}{2}$ for 24 observations.

• Mean: The arithmetic average, $\bar{x} = \frac{\sum x_i}{n}$.

Measures of Dispersion

• Coefficient of Variation (CV): $CV = \frac{\text{Standard deviation}}{\text{Mean}} \times 100\%$

• Standard Deviation: Measures the spread of data around the mean.

Quartiles

• First Quartile (Q1): $Q_1 = \frac{n+1}{4}$th value

• Third Quartile (Q3): $Q_3 = \frac{3(n+1)}{4}$th value

Combinatorics

Counting Principles

• Multiplication Principle: If one event can occur in $n_1$ ways and another in $n_2$ ways, the total number of ways both can occur is $n_1 \times n_2$.

• Permutations with Repetition: For passwords with 3 letters and 2 digits:

  • Number of possible passwords: $n_1 \times n_2$

  • $n_1 = 26^3 = 17,576$ (letters)

  • $n_2 = 10^2 = 100$ (digits)

  • Total: $17,576 \times 90 = 1,581,840$

Discrete Random Variables and Expected Value

Expected Value (Mathematical Expectation)

• Definition: The expected value of a discrete random variable is $E(X) = \sum x_i p(x_i)$.

• Example: For urn draws:

  • Urn #1: $E(X) = (2000 \times 7/15) + (3000 \times 5/15) + (10,000 \times 3/15) = 3933.33$

  • Urn #2: $E(Y) = (3500 \times 2/3) + (5000 \times 1/3) = 4000$

Standard Deviation of a Discrete Random Variable

• Formula: $\text{Var}(Y) = E(Y^2) - [E(Y)]^2$

• Example: $E(Y^2) = (3500^2 \times 2/3) + (5000^2 \times 1/3) = 16,500,000$ $\text{Var}(Y) = 16,500,000 - 16,000,000 = 500,000$

Tables

Probability Distribution Table for Urn Draws

x	2000	3000	10,000
f(x)	7/15	5/15	3/15

y	3500	5000
f(y)	2/3	1/3

Brand Evaluation Table

Brand	Bad	Average	Good	Very Good	Total
Brand 1	7	11	50	40	108
Brand 2	8	37	50	40	135
Brand 3	5	13	50	36	104
Brand 4	2	12	50	36	100
Total	22	73	200	152	447

Summary

• Probability calculations require careful identification of events and their relationships (incompatible, independent).

• The Poisson distribution is useful for modeling event counts over time intervals.

• Measures of central tendency and dispersion (mean, median, quartiles, standard deviation, coefficient of variation) are essential for summarizing data.

• Combinatorics principles help in counting possible arrangements, such as password generation.

• Expected value and variance are key for evaluating random variables in decision-making scenarios.