5.3 and 5.4

Overview of Sections 5.3 and 5.4

  • Discussion of the Binomial Distribution and the Poisson Distribution.

Section 5.3 - The Binomial Distribution

  • The Binomial Distribution relates to repeated trials of an experiment where each trial has two outcomes, termed success and failure.

Key Concepts

  • Trials: Each repetition of the experiment is called a trial.

  • Outcomes: Experiments considered under Binomial Distribution only have two possible outcomes.

Applications

  • Medical Testing: Analyzing the effectiveness of a drug on patients—either effective or not.

  • Sales: In sales pitches, a car salesman either sells the car or does not.

  • Product Testing: A company developing a new soda tests with people who either like or dislike the soda.

Fundamental Elements

To analyze trials in a Binomial Distribution, it is essential to learn about:

  • Factorials

  • Binomial Coefficients

  • Bernoulli Trials

  • The Binomial Distribution itself

Factorials

  • Definition: The factorial of a positive integer k (denoted as k!) is defined as the product of all positive integers up to k.

  • Symbolically, it is expressed as:
    k! = k(k-1)(k-2)…(3)(2)(1)

  • Special case: 0! = 1

  • Example Calculations:

    • 3! = 3 imes 2 imes 1 = 6

    • 4! = 4 imes 3 imes 2 imes 1 = 24

    • 5! = 5 imes 4 imes 3 imes 2 imes 1 = 120

Binomial Coefficients

  • Definition: For a nonnegative integer x (where x ≤ n), the binomial coefficient is defined as:
    inom{n}{x} = rac{n!}{x!(n-x)!}

Bernoulli Trials

  • Trials are classified as Bernoulli trials if they satisfy these conditions:

    1. Each trial has two possible outcomes: success (denoted as s) and failure (denoted as f).

    2. Trials are independent, meaning previous outcomes do not influence subsequent ones.

    3. The probability of success, denoted by p, remains constant across trials.

Binomial Probability Formula

  • The probability distribution of the random variable X (total number of successes in n Bernoulli trials) is given by:
    P(X = x) = inom{n}{x} p^x (1 - p)^{n - x}

  • Here, X is referred to as a binomial random variable, characterized by parameters n (number of trials) and p (probability of success).

  • Valid values for x include 0, 1, 2, …, up to n.

Steps to Calculate Binomial Probability

  1. Identify what constitutes a success.

  2. Determine the success probability (p).

  3. Establish the number of trials (n).

  4. Use the binomial probability formula:
    P(X = x) = inom{n}{x} p^x (1 - p)^{n - x}

Mortality Example

  • According to National Center for Health Statistics, there is an approximately 80% chance that a 20-year-old will be alive at 65.

  • For three randomly selected individuals from this age group, find probabilities as follows:
    a. Exactly two alive, denoted as P(X = 2)
    b. At most one alive, denoted as P(X \le 1)
    c. At least one alive, denoted as P(X \ge 1)

Additional Examples Using Binomial Distribution

  1. Example of rolling a die to show probability distribution.

    • For instance, to find the probability of rolling a '4' exactly once in six rolls:

    • Parameters: n=6, p=1/6 (as per balanced die), x=1

    • Binomial Probability Formula:
      P(X = 1) = inom{6}{1} imes (1/6)^1 imes (5/6)^{5}

Histogram Representation

  • Provides graphical representation of binomial probabilities with different parameter p values (e.g., p = 0.25, p = 0.5, p = 0.75).

Mean and Standard Deviation of Binomial Random Variable

  • The mean BC and standard deviation C3 are expressed as:

  • Mean: BC = n imes p

  • Standard Deviation: C3 = ext{surd}{n imes p(1 - p)}

Section 5.4 - The Poisson Distribution

  • The Poisson Distribution is another discrete probability distribution used in specific scenarios involving frequencies of events within defined intervals.

Applications

  • Common scenarios include:

    • Number of patients arriving at an emergency room during a specific hour.

    • Daily count of telephone calls received at a switchboard.

    • Emission rate of alpha particles from a radioactive substance.

Poisson Probability Formula

  • The probability for a random variable X with a Poisson distribution is given by:
    P(X = x) = rac{e^{-BB} imes BB^x}{x!}

  • Here, BB is a positive real number representing the average rate of events occurring, and e is approximately 2.718.

Mean and Standard Deviation of Poisson Random Variable

  • The mean BC and standard deviation C3 for a Poisson random variable are:

  • Mean: BC = BB

  • Standard Deviation: C3 = ext{surd}{BB}