Probability_and_Distribution_Presentation.pptx

Probability_and_Distribution_Presentation.pptx

• 1.
  Probability and Distribution AnInteractive Teaching Presentation with Infographics and Examples
• 2.
  Definition of Probability •Probability measures how likely an event is to occur. • Formula: P(A) = m/n, where m = favorable outcomes, n = total outcomes. • Range: 0 ≤ P(A) ≤ 1. • Example: Probability of getting a head in a coin toss = 1/2.
• 3.
  Addition Law ofProbability • For any two events A and B: • P(A B) = P(A) + P(B) - P(A ∩ B) ∪ • If mutually exclusive: P(A B) = P(A) + P(B). ∪ • Example: Probability of drawing a red or a king card from a deck.
• 4.
  Multiplication Law ofProbability • P(A ∩ B) = P(A) × P(B|A) • If independent, P(A ∩ B) = P(A) × P(B). • Example: Probability of getting two heads in two coin tosses = 1/4.
• 5.
  Mathematical Expectation • Expectedvalue (mean) of a random variable X: • E(X) = ΣxP(x) • It gives the long-run average of outcomes. • Example: For X = {1,2,3} with P(X)={0.2,0.5,0.3}, E(X)=2.1
• 6.
  Binomial Distribution • Usedwhen trials are independent with two outcomes. • Formula: P(X=r)=C(n,r)p^r q^(n−r) • Mean=np, Variance=npq. • Example: Find P(X=2) when n=5, p=0.4. • Solution: P(X=2)=C(5,2)(0.4)^2(0.6)^3=0.3456
• 7.
  Poisson Distribution • Usedfor rare events. • Formula: P(X=r)=(e^−λ λ^r)/r! • Mean=Variance=λ. • Example: If λ=3, find P(X=2). • Solution: e^−3 × 3²/2! = 0.224
• 8.
  Normal Distribution • Continuous,symmetric, bell-shaped curve. • Formula: f(x)=1/(σ√(2π)) e^{-(x−μ)²/(2σ²)} • Mean=μ, Variance=σ². • Example: Marks ~ N(70,10²), find P(60<X<80) using Z-scores.
• 9.
  Joint, Marginal &Conditional Distributions • Joint: P(X=x, Y=y) • Marginal: P(X=x)=Σ P(X=x, Y=y) • Conditional: P(X=x|Y=y)=P(X=x,Y=y)/P(Y=y). • Example: Table of probabilities for X and Y.
• 10.
  Moment Generating Function (MGF) •MGF: M_X(t)=E(e^{tX}) • Helps find moments: E(X)=M'(0), Var(X)=M''(0)−[M'(0)]². • Example: For Poisson(λ), M_X(t)=exp[λ(e^t−1)].
• 11.
  Characteristic Function • φ_X(t)=E(e^{itX}) •Always exists, useful for Central Limit Theorem proofs.
• 12.
  Data Collection andCompilation • Data collection: Gathering raw information. • Compilation: Organizing and coding data. • Sources: Primary (survey, experiment), Secondary (reports, books).
• 13.
  Classification and Tabulation •Classification: Grouping data (qualitative, quantitative). • Tabulation: Presenting in rows and columns with headings.
• 14.
  Diagrammatic Representation • Visualcomparison of data: Bar diagram, Pie chart, Pictogram. • Each represents proportion or frequency.
• 15.
  Graphical Representation • Linegraph – shows trends over time. • Frequency polygon – connects midpoints. • Ogive – cumulative frequency curve (less than, more than).
• 16.
  Histogram and Ogives •Histogram: Adjacent rectangles with height = frequency. • Ogive: Curve showing cumulative frequencies. • Their intersection gives the median.
• 17.
  Summary Table • Probabilitylaws, Distributions (Binomial, Poisson, Normal), MGF, and Data Presentation summarized for revision.