Stats Chapter 6 Quiz 2

Shape

Same as the probability distribution of X if b > 0.

he sum or difference of independent Normal random variables follows

a Normal distribution.

A linear transformation of a random variable involves

adding or subtracting a constant a, multiplying or dividing by a constant b, or both. We can write a linear transformation of the random variable X in the form Y = a + bX. The shape, center, and spread of the probability distribution of Y are as follows:

when trying to find prob use

normalcdf

Adding a positive constant a to (subtracting a from) a random variable increases (decreases)

the mean of the random variable by a but does not affect its standard deviation or the shape of its probability distribution.

Multiplying (dividing) a random variable by a positive constant b multiplies (divides)

the mean of the random variable by b and the standard deviation by b but does not change the shape of its probability distribution.

If X and Y are independent random variables

then knowing the value of one variable tells you nothing about the value of the other

if X and Y are any two random variables,

μX+Y = μX + μY: The mean of the sum of two random variables is the sum of their means. μX-Y = μX − μY: The mean of the difference of two random variables is the difference of their means.

Center

μY = a + bμX

Spread

σY = |b|σX

In that case, variances add

σ²x+y= σ²x+σ²y: The variance of the sum of two independent random variables is the sum of their variances. σ²x-y= σ²x+σ²y: The variance of the difference of two independent random variables is the sum of their variances.