statistics

• Regression analysis is used to model the relationship between a dependent variable (sales) and an independent variable (number of salespeople).
• The least squares method is used to find the line of best fit that minimizes the sum of squared residuals.
• The slope and intercept of the best-fit line can be interpreted to understand the relationship between the variables.
• Conditional probability is the probability of an event occurring, given that another event has already occurred. It's represented by the notation P(A|B), which is read as "the probability of A given B."

  P(A|B) = P(A and B) / P(B)

  Where:

  • P(A and B) is the probability of both events A and B occurring together.
  • P(B) is the probability of event B occurring.
• Conditional probability is useful in a variety of applications, such as:

  • Medical diagnostics (e.g., the probability of having a disease given a positive test result)
  • Risk assessment (e.g., the probability of an accident given certain weather conditions)
  • Bayesian inference (e.g., updating the probability of a hypothesis given new evidence)
• Bayes' Theorem is a fundamental concept in probability and statistics that describes the relationship between conditional probabilities. It allows us to update the probability of an event occurring based on new information or evidence.

  P(A|B) = (P(B|A) * P(A)) / P(B)

  Where:

  • P(A|B) is the conditional probability of event A given event B
  • P(B|A) is the conditional probability of event B given event A
  • P(A) is the prior probability of event A
  • P(B) is the prior probability of event B
• Bayes' Theorem is particularly useful in situations where you have prior information about the probability of an event, and you want to update that probability based on new evidence or observations. Bayes' Theorem is widely used in fields like machine learning, decision analysis, and medical diagnostics...
• Key probability rules

  • Addition rule
  • Multiplication rule
  • Complementary rule
• Addition rule

  1. P(A or B) = P(A) + P(B) - P(A and B)

  2. This rule states that the probability of A or B occurring is the sum of their individual probabilities, minus the probability of both A and B occurring.
• Multiplication rule

  1. P(A and B) = P(A) * P(B|A)

  2. This rule states that the probability of both A and B occurring is the probability of A occurring multiplied by the conditional probability of B given that A has occurred.
• Complementary rule

  1. P(not A) = 1 - P(A)

  2. This rule states that the probability of an event not occurring is one minus the probability of the event occurring.
•  different types of probability: 1. Probability of an event 2. Probability of a sample 3. Probability of a population
• Probability as a long-run frequency (Empirical Probability):

  • is based on the observed relative frequency of an event over many trials or observations
  • estimated by the proportion of times the event occurs out of the total number of trials
• Model-based (Theoretical) Probability:

  • based on a theoretical model or assumption about the underlying probability distribution
  • assuming that possible outcomes are equally likely, probabilities can be calculated mathematically.
• Personal Probability:

  • reflects an individual's subjective assessment of the likelihood of an event occurring.
  • can be influenced by biases, emotions, and other factors
  • used in decision-making
• Rule 1: When the probability of an event occurring is 0, the event won't occur; if the probability is 1, the event will always occur
• Rule 2

  A: The stock price goes up

  B: The stock price goes down

  C: The stock price remains the same

  When assigning probabilities to these outcomes, we should distribute all of the

  available probability; something always occurs, so the probability of something

  happening is 1: Probability Assignment Rule
• Rule 3: Complement Rule

  Probability getting to class on time is 0.8; probability not getting to class on time

  is 0.2; the set of outcomes that are not in the event A is called the "complement" of A and is denoted A^C

  The probability of an event occurring is 1 minus the probability that it doesn't occur.

  P(A) = 1 - P(A^C)
• Rule 4: The Multiplication Rule

  To find the probability that two independent events occur, we multiply the

  probabilities

  Two independent events A and B, the probability that both A and B occur is the

  product of the probabilities of the two events:

  P(A and B) = P(A) × P(B), provided that A and B are independent
• A random variable is a variable that can take on different values depending on the outcome of a random event.

  For example, consider an insurance company that pays out different amounts based on whether a policyholder dies, becomes disabled, or neither occurs in a given year.
• Expected value of a random variable

  A way to calculate the average value or mean value that we can expect from the random variable
• Calculating expected value


  1. Multiply each possible value of the random variable by its probability of occurring

  2. Sum all those products
• Insurance company example


  • Premium of $500 per year
  • Payout of $100,000 if policyholder dies (probability = 0.001)
  • Payout of $50,000 if policyholder becomes disabled (probability = 0.002)
  • Payout of $0 if neither occurs (probability = 0.997)
• Discrete random variable: a random variable that can list/sum all the outcomes
• Continuous random variable: a random variable that can take on any value
• The standard deviation measures how much the individual values of the random variable deviate from the expected value (mean) on average.
• Steps to calculate the standard deviation of a random variable

  1. Find the deviation of each possible payout from the mean (expected value)

  2. Square each deviation

  3. Take the square root of the variance to get the standard deviation
• Variance


  The expected value of the squared deviations
• Variance is calculated using the formula shown, which involves multiplying each squared deviation by its probability and summing them up
• Standard deviation


  The square root of the variance
• probability


  the likelihood that an event will occur
• probability experiment


  an action, or trial, through which specific results (counts, measurements, or responses) are obtained
• outcome


  the result of a single trail in a probability experiment
• sample space


  the set of all possible outcomes of a probability experiment
• event


  a subset of the sample space. It may consist of one or more outcomes.
• simple event


  an event that consists of a single outcome
• fundamental counting principle


  if one event can occur in "m" ways and a second event can occur in "n" ways the number of ways the two events can occur in sequence is m x n
• Classical probability


  used when each outcome in a sample space is equally likely to occur
• Empirical probability


  based on observations obtained from probability experiment
• Subjective probability


  Result from intuition, education guesses, and estimates