This chapter prepares us to work with distributions of random variables and to find their measures of center and spread.
E[X] = the sum of (x * P(x) for all values of x).
The rules for means are straight-forward. The expected value of a random variable, E[X], is the mean, commonly called mu. The mean of the sum of random variables is the sum of the means. The mean of the difference of random variables is the difference of the means. The E[aX] = a*E[X]. The expected value of a constant is just that constant.
Really complex example: E[aX + bY + c] = a*E[X] + b*E[Y] + c.
When you work with measures of spread you have to be more careful! You cannot add standard deviations. You must work with their squares--the variances.
Var[X] = the sum of ((x-mu)^2 * P(x) for each value of x)
= E[X^2] - (E[X])^2
The Var[aX + b] = a^2 * Var[X]. The constant, b, does not vary, so it contributes NOTHING to the variance.
Now, IF X AND Y ARE INDEPENDENT (THAT"S A BIG IF!!!!!!), then Var(X + Y) = Var(X) + Var(Y). If they are NOT independent, then there is some covariance factor which could be increasing or decreasing the variance. The covariance concept is beyond the scope of this course.
That covariance thing is why we can't calculate the variance of the sum of the math and verbal portions of the SAT directly. We know that these scores are not independent.
Examples from class:
X={1, 11}, Y={-4, 20}, X+Y={-3, 7, 21, 31}
Find the variance of each set and look for a pattern.
Here's another:
X={1, 15}, Y={-4, 44}, X+Y={-3, 11, 45, 59}
Can you create two sets which, when added together, have a variance of 100?
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3 comments:
for part b of the free respose question, is p(at least one negative) = p(neg) + p(neg)^2 + p(neg)^3?
or..is p(at least one neg) = p(only 1 neg) + p(only 2 neg) + p(all three neg)?
if it is the second one, is the p(only 1 neg) = p(neg) * p(not neg)^2? and the p(only 2 neg) = p(neg)^2 * p(not neg)?, etc. because there are three independent events?
ooo wait i might have figured it out...
is p(at least 1 neg) = 1 - p(all not neg)?
which is 1 - p(not neg)^3
is that right?
I guess you're talking about the variance formulas.
When the things being added are independent random variables, treat them like X + Y + Z + etc. and you will get an answer likel n*Var[X]. When you are, instead, multiplying the value of X by a number, use Var[aX] = a^2* Var[X].
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