Statistics Probability and Distribution Concepts

Random Variables

• Discrete Variable: A variable that can be counted. Examples: Number of defective mobile phones in a box, the number of customers at McDonald's.
• Continuous Variable: A variable that can take on any value within a range. Examples: Time spent studying Statistics and Probability, the length of a book, amount of water used to boil an egg.

Probability Distribution

• Properties of Probability Distribution:
  • Sum of all probabilities must equal 1.
  • Probability of each event must be between 0 and 1.
• Mean (Expected Value) Symbol: μ
• Variance Symbol: σ²
• Standard Deviation Symbol: σ

Formuåas

• Mean: μ = Σ[x * P(x)]
• Variance: σ² = Σ[(x - μ)² * P(x)]
• Binomial Distribution: P(x) = (nCx) * p^x * q^(n-x), where:
  • n = number of trials
  • x = number of successes
  • p = probability of success
  • q = probability of failure
  • nCx = number of combinations of n things taken x at a time (n! / (x! * (n-x)!))

Identifying Probability Distributions

• Correct Probability Distributions:
  • A.P(x) = {1/3, 1/3, 1/3}
  • B.P(x) = {0.3, 0.5, 0.1, 0.1}
  • C.P(x) = {0.15, 0.25, 0.3, 0.25, 0.05}
• Incorrect Probability Distribution:
  • D.P(x) = {-0.15, 0.25, 0.3, 0.30} (Because probability cannot be negative)

Outcomes

• Die Toss: The value when a three will appear is C. 3
• Binary (Bernoulli) Distribution: The possible outcomes are D. success and failure.
• Number of Defective Mobile Phones: The possible outcomes are D. {0, 1, 2, 3, 4, 5}

Binomial Distribution Calculation

• n = 25, p = 0.8, find P(x = 15):
  • Use the formula to calculate the probability of exactly 15 successes in 25 trials.

Calculating Mean, Variance, and Standard Deviation

• Car Dealership Problem:
  • Construct a table with columns for x (number of cars sold), P(x), x * P(x), (x - μ)² * P(x).
  • Calculate the mean (μ) using the formula: μ = Σ[x * P(x)].
  • Calculate the variance (σ²) using the formula: σ² = Σ[(x - μ)² * P(x)].
  • Calculate the standard deviation (σ): σ = √σ².

Dice Toss

• Two dice tossed: The probability of getting any combination can be calculated by making a table listing all the possible outcomes. (See the table for detailed solution).

Random Variables

• Discrete: This type of variable can only take on a finite number of values (whole numbers or integers).
• Continuous: This type of variable can take on an infinite number of values within a given range.
• Example:
  • The number of defective mobile phones in a box is a discrete variable.
  • The number of customers at a McDonald's branch is a discrete variable.
  • The time spent studying is a continuous variable.
  • The number of reference books used is a discrete variable.
  • The amount of water used to boil an egg is a continuous variable.
  • The number of flights in a day is a discrete variable.

Probability Distribution

• Two properties:
  • The sum of all probabilities must be equal to 1.
  • Each probability must be between 0 and 1.

Symbols

• Mean: µ
• Variance: σ²
• Standard Deviation: σ

Formulas

• Mean (μ): μ = Σ(x * P(x))
• Variance (σ²): σ² = Σ[(x - μ)² * P(x)]
• Binomial Distribution: P(x) = (nCx) * p^x * q^(n-x)

Binomial Distributions

• Outcomes: Success and failure.
• Probability of success: p
• Probability of failure: q = 1 - p

Example: Defective Phones

• The possible number of defective phones in a box of five is: {0, 1, 2, 3, 4, 5}
• The binomial distribution with n=25 and p=0.8 represents the probability of getting a certain number of successes (x) in 25 trials where the probability of success is 0.8.

Car Dealership Example

• The mean is calculated by summing the product of each number of cars sold and its corresponding probability.
• The variance is calculated by summing the product of the squared difference between each number of cars sold and the mean, and its corresponding probability.
• The standard deviation is the square root of the variance.

Two Dice Example

• The possible outcomes of rolling two dice can be represented using a table, showing the sum of the values on each die.