Statistics 106: Ch. 7 Flashcards

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.