COMP1321 Mathematics II
Joint Probability Distributions
A joint probability distribution describes the probabilities of two or more random variables occurring simultaneously.
Recap: Joint Probability Distributions
- Definition: A joint probability distribution function, , is defined for discrete random variables and .
- Properties:
- for all possible values of and .
- .
- Continuous Variables: For continuous random variables and , their joint probability density function (JPDF) is denoted by .
- .
- .
- Marginal Probability Distributions: These are obtained by summing or integrating the joint probability distribution over the other variable.
- Discrete:
- Continuous:
Covariance
Covariance measures how two random variables change together. A positive covariance indicates that the variables tend to increase or decrease together. A negative covariance indicates that one variable tends to increase when the other decreases. A covariance of zero suggests no linear relationship.
- Formula:
- Calculation Steps:
- Calculate using the marginal distribution of .
- Calculate using the marginal distribution of .
- Calculate using the joint probability distribution:
- Compute the covariance: .
Example: Covariance from Empirical Data
Given the following JPDF:
P(X,Y) | P(Y) | |||
0.10 | 0.08 | 0.02 | 0.20 | |
0.05 | 0.03 | 0.02 | 0.10 | |
0.07 | 0.08 | 0.15 | 0.30 | |
0.01 | 0.30 | 0.09 | 0.40 | |
P(X) | 0.23 | 0.49 | 0.28 | 1.00 |
- Calculate
- Calculate :
- Calculate :
- Calculate Covariance:
Correlation
Correlation is a standardized measure of covariance. It indicates the strength and direction of a linear relationship between two variables. The correlation coefficient ranges from -1 to +1 .
- Formula:
where and are the standard deviations of and , respectively.
- Calculation Steps:
- Calculate the covariance, .
- Calculate the variance of .
- Calculate the standard deviation of , .
- Calculate the variance of .
- Calculate the standard deviation of .
- Compute the correlation: .
Example: Correlation from Empirical Data
Using the JPDF and previously calculated values from the covariance example: , .
- Calculate : .
- Calculate
- Calculate
- Calculate :
- Calculate
- Calculate
- Calculate Correlation:
Measures of Dispersion
Measures of dispersion quantify the spread or variability of data points in a probability distribution.
Variance
Variance is the average of the squared differences from the mean. It measures how spread out the data is.
- Formulas:
where is the mean.
- Calculation Steps (using ):
- Calculate the mean, .
- Calculate the second moment, .
- Compute the variance: .
Standard Deviation
Standard deviation is the square root of the variance. It provides a measure of spread in the same units as the data.
- Formula:
- Interpretation: represents an interval that typically contains about of the data in a normal distribution.
Skewness
Skewness measures the asymmetry of a probability distribution.
- Formula:
where is the mean, is the median, and is the standard deviation.
- Interpretation:
- Positive Skew: The tail of the distribution is longer on the right side. The mean is typically greater than the median.
- Negative Skew: The tail of the distribution is longer on the left side. The mean is typically less than the median.
- Zero Skew: The distribution is symmetric (e.g., a normal distribution). The mean and median are equal.
The Normal Distribution
The normal distribution, often called the “bell curve,” is a continuous probability distribution that is symmetric about its mean.
- Notation: , where is the mean and is the variance.
- Probability Density Function (PDF):
- Properties:
- The mean, median, and mode are all equal to .
- The curve is symmetric around the mean.
- The total area under the curve is 1.
- A linear transformation of a normally distributed random variable is also normally distributed. If and , then .
- The sum of independent normally distributed random variables is also normally distributed. If and , then .
Standard Normal Distribution
The standard normal distribution is a special case of the normal distribution where the mean is 0 and the variance is 1 ( ).
- Notation: .
- PDF:
Cumulative Distribution Function (CDF) and Z-Score Table
The CDF of a normal distribution, , gives the probability that a random variable takes on a value less than or equal to a specific value. For the standard normal distribution, this is .
- Transformation: To find probabilities for a general normal distribution , we standardize the variable to :
- Calculating Probabilities:
- Z-Score Table: A Z-score table provides pre-calculated values of for the standard normal distribution.
Example: Calculating Probability with Normal Distribution
Suppose the heights of British teenagers are normally distributed with a mean and a standard deviation . What percentage of British late teens are between 160 cm and 180 cm tall?
- Identify parameters: .
- Define the range: .
- Standardize the values:
。
。
4. Use the Z-score table to find probabilities:
- Calculate the probability for the range:
Therefore, approximately of British late teens are between 160 cm and 180 cm tall.