Probability Chapter 5

Probability Formulas

  • Binomial Probability Formula:

    • Used for discrete variables.

  • Exponential and Poisson Distributions:

    • Exponential is often not based on specific values for x.
    • Poisson distribution involves different functions based on the scenario.

  • Negative Binomial Distribution:

    • Also has its own functions.

Continuous vs Discrete Probability

  • In continuous probability:
    • No specific value for x; x can take a range of values.
  • To find the probability for a continuous variable:
    • You integrate over the range rather than summing points.
    • Example: If asked for the probability that x is between 2 and 5:
    • Discrete Case: you sum the probabilities at 2, 3, and 4 (5 is not included).
    • Continuous Case: integrate the probability density function.

Probability Density Function (PDF)

  • Definition of PDF (f(x)):
    • Describes the likelihood of a random variable falling within a particular range of values.
    • The function must satisfy:
    • The total area under the curve (for the entire range) equals 1.
    • For any range [a, b]:
    • Integral of f(x) dx from a to b gives the probability that x falls within that interval:

P(a<x<b)=∫abf(x) dxP(a < x < b) = \int_a^b f(x) \,dx

  • Key Concept: The PDF gives the shape of the probability distribution but not direct probabilities.

Example of Density Function Calculation

  • If integrating from minus infinity to plus infinity for a given function:

    • Must equal 1 (total probability).

  • Finding the PDF:

    • Example: If x represents temperature with an unknown initial equation, one example given involves:
    • Finding x between 30 and 90 degrees.
    • Can integrate the density function over this range to find total area (i.e., probability).

Integrating Probability Density Functions (PDF)

  • When asked for probability at a specific point (e.g., x = 50):
    • The probability at any specific point is zero due to no area (width = 0).
    • Thus, the PDF is not defined at a point.

Limit Concept in Continuous Probability

  • Although the point probability is zero, using limits can give a sense of probability at boundaries.

    extIntegral:∫leftrightf(x) dxext{Integral: } \int_{left}^{right} f(x) \, dx

  • In the context of arrival times related to a train example:

    • Analogous to assessing probabilities at discrete points versus continuous limits.

Cumulative Distribution Function (CDF)

  • Definition of CDF:

    • The cumulative probability from the left of -infinity to a value x:
    • F(x)=P(X≤x)=∫−extinfinityxf(t) dtF(x) = P(X \leq x) = \int_{- ext{infinity}}^x f(t) \, dt

  • Relationship between PDF and CDF:

    • If differentiating the CDF, you'll recover the PDF:

dF(x)dx=f(x)\frac{dF(x)}{dx} = f(x)

Uniform Distribution

  • Definition: A type of distribution where every value in a certain interval has the same probability.
  • Density Function for Uniform Distribution:
    • Given by:
      f(x)=1β−αf(x) = \frac{1}{\beta - \alpha}
  • Where x lies between [α, β].
    • This ensures the total area under the curve equals 1 (total probability)

Parameters of Uniform Distribution

  • For a uniform distribution:
    • The parameters are α and β, defining bounds where the variable has equal probability.
  • Constant Value:
    • Integrating the density function should yield the area of 1 to keep the integrity of distribution.

Random Variable Properties

  • Continuous Random Variable created from transformations; when using CDF,

    • Deriving mean/variance depend upon parameters of distributions:
    • Example: For uniformly distributed random variable between [7, 8], mean and variance can be derived.

  • Scenarios involving variables leading to outcomes determine the overall probability distributions being analyzed.

Standard Normal Distribution

  • Various types of transformations, particularly to achieve a mean of 0 and standard deviation of 1. This is critical for probability table clearances.

    • For Z-scores, use:
      Z=X−μσZ = \frac{X - \mu}{\sigma}

  • Understanding bell curves and integration leads towards recognizing behavior of distributions.

The Significance of Standard Deviation

  • Standard deviation defines spread: Indicates how data varies from the mean.
    • Knowing these provides a framework for analyzing probabilities over ranges.

Example of Finding Z-Scores

  • If X equals 8 in a distribution where \mu = 5 and \sigma = 2:
    • Convert via:

Z=8−52=1.5Z = \frac{8 - 5}{2} = 1.5

  • Use Z-score in the CDF for finding probabilities based on this normal distribution framework.