Statistics 106: Ch. 7 Flashcards

Questions and Answers

What is a Uniform Probability Distribution?

A distribution where one interval is more likely than another.

A type of distribution that only allows integer outcomes.

A distribution that is always symmetrical.

A distribution where any two intervals of equal length are equally likely. (correct)

The probability of observing one particular value for a Continuous Random Variable is greater than 0.

False

What satisfies the properties of a Probability Density Function?

The total area under the graph must equal one and the height must be non-negative.

Match the following key terms with their definitions:

Probability Density Function = An equation used to compute probabilities of continuous random variables. Model = An equation, table, or graph used to describe reality. Normal Curve = A curve used to describe continuous random variables that are normally distributed. Standard Normal Random Variable = A variable with a mean of 0 and a standard deviation of 1. Signup and view all the answers

What does the area under the graph of a Probability Density Function represent?

The probability of observing a value of the random variable in that interval. Signup and view all the answers

The __________ of a distribution is the number that appears the most frequently.

mode Signup and view all the answers

What are the inflection points on the normal curve?

Mean - 1 standard deviation and Mean + 1 standard deviation. Signup and view all the answers

Changing the standard deviation affects the center of the normal distribution.

False Signup and view all the answers

What is true about the properties of the Normal Density Curve?

It is symmetric about the mean. Signup and view all the answers

What does a Z-Score allow us to do?

Transform the random variable into a standard normal variable. Signup and view all the answers

What is the formula for standardizing a normal random variable?

Z = (X - mean) / (standard deviation). Signup and view all the answers

Study Notes

Probability Distributions

A uniform probability distribution occurs when two intervals of equal length have equal probabilities.
Continuous random variables (CRVs) can take infinite outcomes, leading to a probability of 0 for any specific value.

Probability Density Function (PDF)

A PDF computes probabilities for CRVs and must satisfy:
- Total area under the curve equals one.
- Height of the curve is non-negative.
The area under a PDF over an interval indicates the probability of the variable falling within that interval.

Normal Distribution

The normal curve represents normally distributed continuous random variables, characterized by a bell shape.
A random variable is normally distributed if its frequency distribution resembles a normal curve.

Characteristics of Normal Curves

The mode represents the highest point (most frequent value), median splits data into two halves, and mean is the balancing point.
For symmetrical single-peaked curves, mean, median, and mode are equal.

Inflection Points

Inflection points occur at one standard deviation above and below the mean on a normal curve, indicating curvature changes.

Effects of Mean and Standard Deviation

Changing the mean shifts the curve left or right without altering its shape.
Altering standard deviation affects slope steepness while maintaining the center of the graph.

Properties of the Normal Density Curve

Symmetry about the mean is a key property.
The curve peaks at the mean, where mean = median = mode.
Inflection points occur at one standard deviation from the mean.
Total area under the curve equals one, with equal areas on both sides of the mean.
Empirical rule:
- 68% of data within one standard deviation.
- 95% within two standard deviations.
- Approximately 99.7% within three standard deviations.

Z-Score Transformation

Z-scores allow for transforming a normally distributed variable into one with mean 0 and standard deviation 1.
Standardizing a variable uses the formula Z = (X - mean) / standard deviation.

Standard Normal Distribution

A standard normal random variable has a mean of 0 and a standard deviation of 1, often achieved through standardization.

Description

Test your knowledge on the properties of the normal distribution with these flashcards. Learn key concepts such as uniform probability distribution and continuous random variables. Perfect for those studying Statistics 106.