Statistical Measures for Random Variables

Powerpoint only, not Including WebAssign

The expected value of a random variable X is

the mean or average value of X over all possible outcomes.

It is the weighted average of all possible values that X could have, weighted by the probabilities of these values to occur

The standard deviation of a random variable is its

long run average deviation from the mean

The variance of a random variable is the

average squared deviation from the mean, with each deviation weighted by its probability to occur.

To get variance, calculate the squared deviations of each possibly outcome, and multiple by its probability, then add.

The law of large numbers states _

as sample size increases, the sample mean will converge to the mean of the population. Thus, larger sample sizes are guaranteed to produce results that are close to the population, while smaller sample sizes might have a mean that is considerably different

Key differences in Probability Distribution vs. Empirical Distribution

Probability Distribution - Theoretical model or mathematical framework that provides complete information about all probabilities. This is constant, and is based on theoretical assumptions.

Empirical distributions are observed data from experiments or observations. This is based on sample data and can only approximate the true distribution. This is not constant, and will change when collecting different data sets.

When we have two variables, we would like to know how strongly they are related. This is measured by

correlation, which measures the strength of the linear relationship between X and Y

Correlation is scaled between

1 and -1.

The smaller the scatter of points around a straight line, the smaller the prediction errors for Y, and the closer the correlation to 1 or -1.

If a correlation has a positive sign _

the trendline has an upward slope, and Y tends to increase when X increases

If correlation has a negative sign _

the trendline has a downward slope and Y tends to decrease when X increases

Correlation is only valid for _ trends

LINEAR.

The correlation measures how close the points are from a straight line, regardless if the data follow a straight line or now. ONLY MEANINGFUL FOR LINEAR TRENDS

Covariance

The covariance measures the degree to which two random variables move in the same or opposite directions.

A positive covariance indicates

the variables tend to move in the same direction

A negative covariance indicates

the variables tend to move in opposite directions.

If covariance is equal to 0, this implies

X and Y move independently of each other

T / F : The covariance and correlation always have the same sign

TRUE!!

Difference between covariance and correlation

Covariance doesn’t indicate the strength of the relationship, ONLY THE DIRECTION!!

The strength is measured with correlation.