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
Slide 1 Topic: Probability - Revision of Random Variable Definition of random variable Discrete random variable Continuous random variable Examples of random variables