Normal Distribution: Properties and Applications

Chapter 7: Normal Distribution

Section 7.1 - Properties of the Normal Distribution

Continuous Probability Distributions

Continuous probability distributions describe the probabilities of outcomes for continuous random variables. One important example is the uniform probability distribution, also known as the rectangular distribution, where all outcomes within a given interval are equally likely.

• Probability Density Function (pdf): The probability of a continuous random variable is represented by the area under its curve, called the probability density function (pdf), denoted as f(x).

• Total Area: The total area under the curve for all possible values of the random variable equals 1.

• Function Representation: The function f(x) corresponds to the graph of the distribution.

Cumulative Distribution Function (cdf)

The cumulative distribution function (cdf) gives the probability that a random variable is less than or equal to a certain value. It is used to evaluate probability as area under the curve up to a specific point.

• Interval Probabilities: Probability is found for intervals of x values, not for individual values. For example, $P(c < x < d)$ is the probability that the random variable X is between c and d.

• Area Interpretation: This probability is the area under the curve above the x-axis, between c and d.

• Single Value Probability: $P(x = c) = 0$ for continuous distributions, since the area at a single point is zero.

Inequalities in Probability

Common phrases are used to indicate inequalities in probability statements. The following table summarizes these:

Symbol	Phrases
>	Is greater than, more than, larger than
≥	Is greater than or equal to, is at least, is the minimum
<	Is less than, is smaller than, has fewer than
≤	Is less than or equal to, is at most, is the maximum

The Normal Distribution

The normal distribution is the most important continuous probability distribution in statistics. It is widely used in many disciplines and is characterized by its bell-shaped curve.

• Symmetry: The curve is symmetrical about a vertical line through the mean.

• Total Area: The area under the curve equals one.

• Parameters: The normal distribution is defined by two parameters: the mean ($bc$) and the standard deviation ($c$).

• Notation: If $X$ is normally distributed with mean $bc$ and standard deviation $c$, we write $X hicksim N(bc, c)$.

• Cumulative Distribution Function: The cdf, $F(x)$, gives the probability that $X$ is less than or equal to $x$.

The Standard Normal Distribution and Z-Scores

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Values from any normal distribution can be converted to z-scores, which indicate how many standard deviations a value is from the mean.

• Z-Score Formula: $z = rac{x - bc}{c}$

• Interpretation:

  • Values of $x$ larger than the mean have positive z-scores.

  • Values of $x$ smaller than the mean have negative z-scores.

  • If $x$ equals the mean, then $z = 0$.

Empirical Rule

The Empirical Rule describes the percentage of data within certain numbers of standard deviations from the mean in a bell-shaped (normal) distribution:

• Approximately 68% of data falls within 1 standard deviation ($bc c$ to $bc + c$).

• Approximately 95% of data falls within 2 standard deviations ($bc - 2c$ to $bc + 2c$).

• Approximately 99.7% of data falls within 3 standard deviations ($bc - 3c$ to $bc + 3c$).

Section 7.2 - Using the Normal Distribution

Calculating Probabilities with the Normal Distribution

Probabilities in the normal distribution are represented by areas under the curve. These areas can be calculated using the cumulative distribution function or standard normal tables.

• Left-Tail Probability: $P(X < x)$ is the area to the left of the vertical line through $x$.

• Right-Tail Probability: $P(X > x) = 1 - P(X < x)$ is the area to the right of the vertical line through $x$.

• Calculators and Tables: Probabilities are often calculated using statistical calculators or standard normal tables.

Visual Representation

Shaded regions under the normal curve represent probabilities for intervals or specific conditions (e.g., $P(X < x)$ or $P(X > x)$).

• Purple Region: Area to the left of $x$ (for $P(X < x)$).

• White Region: Area to the right of $x$ (for $P(X > x)$).

Summary Table: Probability Regions

Probability Statement	Region Under Curve
$P(X < x)$	Left of $x$
$P(X > x)$	Right of $x$
$P(c < X < d)$	Between $c$ and $d$

Additional info: The notes infer standard notation and usage for normal distributions, including the use of calculators and tables for probability calculations, and the interpretation of z-scores for standardization.