Normal Distribution Study Notes

Overview of Continuous Probability Distributions

• Continuous probability distributions differ from discrete ones in that continuous variables have infinitely many possible values.

• Example of continuous variables includes measurements such as height, weight, and temperature.

• The probability that a continuous random variable $X$ takes on any single, exact value is always equal to 0, known as "point probability."

• Instead of focusing on the probabilities of individual points, we concentrate on intervals of values.

• To find these probabilities, we utilize a probability distribution function (pdf).

• Calculus is often involved in the calculations of such probabilities.

Probability Density Functions (PDFs)

• A probability density function (pdf) is defined as an equation that computes probabilities for continuous random variables.

• Requirements for a PDF: - It must have a total area under the graph of 1 for all possible values of $X$, commonly referred to as the "area under the curve" (AUC).

  • The height of the PDF must be at least 0 for all possible values of $X$ (the pdf cannot be negative).

Examples of PDFs

• Graphs depicting various probability density functions can adopt many shapes.

• Some representations may include:- Uniform distributions, where height and probabilities are constant.

  • Example: Uniform distribution on interval $(0, 5)$ where the total area is 1.

  • Other mathematical expressions that outline specific PDFs in various contexts.

The Uniform Distribution

• The uniform distribution represents the simplest case of a continuous distribution.

• In this distribution, all intervals of equal length have equal probabilities due to its rectangular shape.

• Height Determination:- The height of the rectangle is calculated based on the total area under the section of the curve.

Finding Probabilities

• In continuous distributions like the uniform distribution, the probability that $X$ falls within a certain interval is equivalent to the area under the curve of the PDF for that interval.

• For the uniform distribution, the area of the rectangle defines this probability.

• Formula for Area of a Rectangle:- Area = Base $\times$ Height.

Example: Uniform Distribution Calculation

• Given a uniform distribution on the interval $(0, 5)$. Since the total area must be 1, the height of the distribution is $1/(5-0) = 1/5 = 0.2$.

  • To find P(X > 3):

    1. Identify the interval: We are interested in the interval $(3, 5)$.

    2. Determine the base: The base of this interval is $5 - 3 = 2$.

    3. Apply the height: The height of the uniform distribution is $0.2$.

    4. Calculate the area (probability): Area = Base $\times$ Height $= 2 \times 0.2 = 0.4$. Thus, P(X > 3) = 0.4.

  • To find P(X < 3), or P(0 < X < 3), since the distribution starts at 0:

    1. Identify the interval: We are interested in the interval $(0, 3)$.

    2. Determine the base: The base of this interval is $3 - 0 = 3$.

    3. Apply the height: The height of the uniform distribution is $0.2$.

    4. Calculate the area (probability): Area = Base $\times$ Height $= 3 \times 0.2 = 0.6$. Thus, P(X < 3) = 0.6.

Moving to More Complex Distributions

• As we explore more complex distributions, it remains crucial to recognize that the probability of $X$ falling within the interval $(a, b)$ is still determined by the area under the PDF between $a$ and $b$.

• Distribution shapes can vary widely, but the principle remains.

The Normal Distribution Characteristics

• The normal distribution is characterized by several key properties:- The mean, median, and mode are all equal.

  • The distribution is perfectly symmetrical around the mean.

  • The total area under the curve equals 1.

  • Inflection points exist exactly 1 standard deviation away from the mean.

  • The normal curve never touches or crosses the x-axis.

Impacts of Changing the Mean and Standard Deviation

• Altering the mean ($\mu$) and the standard deviation ($\sigma$) of the normal distribution affects its shape and positioning.

Review: The Empirical Rule

• The empirical rule outlines how data in a normal distribution behaves in relation to standard deviations:- Approximately 68% of data falls within 1 standard deviation of the mean ($\mu \pm \sigma$).

  • About 95% falls within 2 standard deviations ($\mu \pm 2\sigma$).

  • Approximately 99.7% falls within 3 standard deviations ($\mu \pm 3\sigma$).

  • Visual representation of probabilities associated with these ranges, often displayed in segmented curves.

Area Under the Curve

• The area under the curve can represent different concepts:- Probability: Finds the likelihood of an event occurring within a specific range.

  • Proportion of Population: Relates to how much of a total population falls within a defined interval.

  • Percentile: Indicates the percentage of scores that fall below a certain value.

Example Analysis Task

• Given the graph of a normal distribution:- Identify the mean ($\mu$) and standard deviation ($\sigma$) from the graph.

  • Analyze and state what an area under the curve to the right of a certain point ($X=10$, area=0.0668) represents in terms of probability, proportion, and percentile.

Moving Forward

• The remainder of Chapter 7 will delve deeper into the applications of the normal distribution, covering:- Calculation of z-scores.

  • Techniques for calculating probabilities.

  • Finding percentiles within data distributions.

  • Assessing whether data follows a normal distribution through various statistical tests.