Slide 1

  • Topic: Probability - Revision of Random Variable
  • Definition of random variable
  • Discrete random variable
  • Continuous random variable
  • Examples of random variables

Slide 2

  • Probability mass function (PMF)
  • Definition of PMF
  • Properties of PMF
  • Examples of PMFs
  • PMF for discrete random variables

Slide 3

  • Probability density function (PDF)
  • Definition of PDF
  • Properties of PDF
  • Examples of PDFs
  • PDF for continuous random variables

Slide 4

  • Cumulative distribution function (CDF)
  • Definition of CDF
  • Properties of CDF
  • Examples of CDFs
  • CDF for both discrete and continuous random variables

Slide 5

  • Expected value of a random variable
  • Definition of expected value
  • Properties of expected value
  • Calculation of expected value
  • Examples of expected value calculations

Slide 6

  • Variance of a random variable
  • Definition of variance
  • Properties of variance
  • Calculation of variance
  • Examples of variance calculations

Slide 7

  • Standard deviation of a random variable
  • Definition of standard deviation
  • Properties of standard deviation
  • Calculation of standard deviation
  • Examples of standard deviation calculations

Slide 8

  • Joint probability distribution
  • Definition of joint distribution
  • Joint probability mass function (Joint PMF)
  • Joint probability density function (Joint PDF)
  • Examples of joint distributions

Slide 9

  • Marginal probability distribution
  • Definition of marginal distribution
  • Calculation of marginal distribution from joint distribution
  • Examples of marginal distributions
  • Marginal PMF and marginal PDF

Slide 10

  • Conditional probability distribution
  • Definition of conditional distribution
  • Calculation of conditional distribution from joint distribution
  • Examples of conditional distributions
  • Conditional PMF and conditional PDF
  • Probability mass function for multiple random variables
  • Joint PMF for discrete random variables
  • Example: Coin tossing experiment with two coins
  • Joint PMF table
  • Marginal PMF from joint PMF
  • Joint probability distribution for continuous random variables
  • Joint PDF for continuous random variables
  • Example: Height and weight of individuals
  • Joint PDF graph
  • Marginal PDF from joint PDF
  • Conditional probability mass function
  • Definition and formula for conditional PMF
  • Example: Conditional probability of getting a head given that the first coin is a tail
  • Calculation of conditional PMF
  • Conditional PMF table
  • Conditional probability density function
  • Definition and formula for conditional PDF
  • Example: Conditional probability of weight given a certain height
  • Calculation of conditional PDF
  • Conditional PDF graph
  • Functions of random variables
  • Transformation of random variables
  • Example: Transformation of a random variable using a linear function
  • Calculating the transformed random variable
  • Properties of transformed random variables
  • Moment generating function (MGF)
  • Definition and formula for MGF
  • Example: Calculating MGF for a discrete random variable
  • Calculation of MGF using the formula
  • Properties and uses of MGF
  • Sum of random variables
  • Properties of the sum of random variables
  • Example: Sum of two dice rolls
  • Calculation of the sum of two random variables
  • Probability distribution of the sum
  • Product of random variables
  • Properties of the product of random variables
  • Example: Product of two independent random variables
  • Calculation of the product of two random variables
  • Probability distribution of the product
  • Moment generating function of the sum of random variables
  • Calculation of MGF for the sum of random variables
  • Example: Calculating MGF for the sum of two dice rolls
  • Calculation of MGF using the MGFs of individual random variables
  • Properties and uses of MGF for the sum of random variables
  • Central limit theorem
  • Statement of the central limit theorem
  • Example: Rolling a fair die multiple times
  • Illustration of the central limit theorem with histogram
  • Implications and applications of the central limit theorem

Slide 21

  • Moment generating function of the product of random variables
  • Calculation of MGF for the product of random variables
  • Example: Calculating MGF for the product of two independent random variables
  • Calculation of MGF using the MGFs of individual random variables
  • Properties and uses of MGF for the product of random variables

Slide 22

  • Covariance of random variables
  • Definition of covariance
  • Calculation of covariance
  • Example: Calculating covariance between two random variables
  • Properties and interpretation of covariance

Slide 23

  • Correlation coefficient of random variables
  • Definition of correlation coefficient
  • Calculation of correlation coefficient
  • Example: Calculating correlation coefficient between two random variables
  • Properties and interpretation of correlation coefficient

Slide 24

  • Properties of independent random variables
  • Definition of independence
  • Calculation of independence using joint distribution
  • Example: Checking independence between two random variables
  • Properties and implications of independence

Slide 25

  • Properties of dependent random variables
  • Definition of dependence
  • Calculation of dependence using joint distribution
  • Example: Checking dependence between two random variables
  • Properties and implications of dependence

Slide 26

  • Law of Large Numbers
  • Statement of the Law of Large Numbers
  • Example: Tossing a fair coin multiple times
  • Illustration of the Law of Large Numbers with histograms
  • Implications and applications of the Law of Large Numbers

Slide 27

  • Central Limit Theorem for sums of independent random variables
  • Statement of the Central Limit Theorem for sums
  • Example: Sum of independent, identically distributed random variables
  • Calculation of the mean and variance of the sum
  • Implications and applications of the Central Limit Theorem for sums

Slide 28

  • Central Limit Theorem for averages of independent random variables
  • Statement of the Central Limit Theorem for averages
  • Example: Average of independent, identically distributed random variables
  • Calculation of the mean and variance of the average
  • Implications and applications of the Central Limit Theorem for averages

Slide 29

  • Limiting Distribution
  • Definition of limiting distribution
  • Examples: Binomial distribution converging to Poisson distribution
  • Calculation of the limiting distribution
  • Properties and interpretations of limiting distributions

Slide 30

  • Applications of probability and random variables in real-life scenarios
  • Example: Probability in finance and stock market analysis
  • Example: Probability in healthcare and medical research
  • Example: Probability in sports and gaming
  • Example: Probability in weather forecasting and climate modeling
  • Importance and relevance of studying probability and random variables for various fields