Stats Test 4 (6.2 Transforming and Combining Random Variables)
effect on a random variable of adding (or subtracting) a constant
Adding the same number a (which could be negative) to each value of a random variable: 1. Adds a to measures of center and location (mean, median, quartiles, percentiles). 2. Does not change measures of spread (range, IQR, standard deviation). 3. Does not change the shape of the distribution.
variance of the difference of random variables
For any two independent random variables X and Y, if D = X - Y, then the variance of D is σ²D = σ²x - σ²y
variance of the sum of random variables
For any two independent random variables X and Y, if T = X + Y, then the variance of T is σ2T = σ2x + σ2y. 1. In general, the variance of the sum of several independent random variables is the sum of their variances. 2. Remember that you can add variances only if the two random variables are independent, and that you can NEVER add standard deviations.
mean of the difference of random variables
For any two random variables X and Y, if D = X − Y, then the expected value of D is E(D) = μD = μX − μY
mean of the sum of random variables
For any two random variables X and Y, if T = X + Y, then the expected value of T is E(T) = μT = μX + μY In general, the mean of the sum of several random variables is the sum of their means.
effect of a linear transformation on the mean and standard deviation
If Y = a + bX is a linear transformation of the random variable X, then 1. The probability distribution of Y has the same shape as the probability distribution of X. 2. μy = a + bμx. 3. σY = ⎜b⎜σx (since b could be a negative number).
independent random variables
If knowing whether any event involving X alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then X and Y are ...
Effect on a Random Variable of Multiplying (or Dividing) by a Constant
Multiplying (or dividing) each value of a random variable by a number b: 1. Multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b. 2. Multiplies (divides) measures of spread (range, IQR, standard deviation) by ⎜b⎜. 3. Does not change the shape of the distribution.
