Continuing from here on Wednesday:
Go thru handout, as we work through more results of Algebra of expectations.
If X and Y are independent, then E(X · Y) = E(X)
· E(Y) , I'll prove it. Accept
it for HW tonight..
New this semester:
VarX = E(X2) - (E(X))2
(We can turn this around and find E(X2) from VarX and E(X))
I'll prove it. Accept it for HW tonight..
Cov(X, Y) = E[(X-E(X)) ·
(Y-E(Y))] (def.) = E(X ·
Y) - E(X) · E(Y)
Var(X+Y) = Var(X) + Var(Y) + 2Cov(X, Y)
(a cov term for every pair, if summing more than 2)
If
X and Y are independent, Cov(X,Y) = 0, and we get our familiar Var sum.
Correlation "rho"of X, Y is Cov "standardized" by
dividing by both standard deviations (p. 235).
is Theoretical version of correlation coefficient r.
I'll go over variance of Hypergeometric.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Read Ash pp. 220-235 (ignore anything
with integral signs) Especially note proof of "More practical way
to find VarX, p. 225. It should be understandable.
HW: Many are taken from list on
bottom of Algebra handout; repeated here. Eventually you'll do them
all.
Due Wed:
Derivatives review:
A) Find the derivative with respect to
w:
(w3- h)/(3+w + 2w2),
exp(-w2), ln(2 + 3w2
)
Find the second derivative with respect to q:
q, q2, q3, q4, q5, q6,
qx
B) Graph the four points in the x,y plane (0,2), (1,0), (1,4), (2,2).
Let each point be equally likely (prob = 1/4). Let X be the value
on the x-coordinate, and Y be the value on the Y coordinate.
a) Find E(X + Y) by evaluating x+y for each of the 4 points,
multiplying by the probability, and summing.
b) Find E(XY) by evaluating xy for each of the 4 points, multiplying
by the probability, and summing.
c) Find the probability distribution of X. Find the probability
distribution of Y. Find E(X). Find E(Y).
d) Check if E (X+Y) = E(X) + E(Y).
e) Check if E(X · Y) = E(X) ·
E(Y)
f) Check if X and Y are independent. (For independence,
for any x,y pair, P(x,y) = P(X=x) · P(Y=y). So to show
non-independence, you only need to find one x,y for which that is not
true.)
p. 223, #14. (We haven't yet proved that If X and Y are independent, then E(X · Y) = E(X) · E(Y), but we've talked about it, and it's true. Your results from problem B may be helpful here also.)
p. 233, Accept the alternate formula for variance: VarX = E(X2)
- (E(X))2 (formula and proof,
p. 225) and use it where useful.
#1, #3
#6 a. We'll work toward proving it next time.
# 7, using the rules you learned from Moore
& McCabe, & VarX = E(X2)
- (E(X))2
#16 (they mean: you know the distribution of X if
you think about it.)
D) On a separate page, find E(X(X-1)) for the Poisson distribution. Keep it for next time. (Make a table: the beginning is shown on the bottom of the Algebra handout. Then you can use the trick on p. 76)
E) Use the algebra of Expected values to get a relationship among E(X(X-1)), E(X2), and E(X)
Continue to fill out your Named distribution sheet. You've now
proved or should have read the proofs for:
Binomial (Mean and Variance)
Geometric (Mean: 3 ways--2 in class, 1 on p. 81)
Hypergeometric (Mean, p. 80--same as Binomial. Checked
in an example)
Poisson (Mean: in class, also p. 76)
Due Monday after break? more....
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