Quiz postponed to Wednesday: Closed
book, Expected value stuff, chosen from:
Deriving E of Poisson, Var of Poisson from E(X(X-1)) and E(X),
Proofs for Var and Cov alternate formulas. Var(X+Y) = (derivation)
Independence and E(XY), Cov(X,Y)
HW questions?
4.2: Cumulative Distribution Function
(CDF,
"Distribution Function")
F(x) = P(X < x).
Defined for every x on the real line. Capital letter.
Often what tables table.
Discrete: Sum of probabilities to the
left of x (including x). Jumps at each lump. "Step
function."
See Poisson table (Old handout,
or get new)
P (X < a) = F(a), P (X <
b) = F(b), so
P (a < X < b) = F(b) - F(a)
(note missing = at left end) Must watch ends carefully
For lambda = 4: P(X< 3)
= F(3) = .433 0___1___2___3___4___5___6___7___
P(X > 3) = 1 - F(3) = .567 = P(X > 4)
P(4 < X < 6) = P(3 < X < 6) = F(6) - F(3)
= .889-.433
CDF from probability function: F(x) =
step function (graphed in class)
x 1 3
4 6
0, x < 1
p .2 .2 .3 .3
.2, 1< x < 3
.4, 3< x < 4
.7, 4< x < 6
1, 6< x
Continuous: Area to the left of x under
the density. Continuous function.
(Some) Properties: F(x) --> 1 as x -->
infinity (it may actually get to 1, earlier)
F(x) --> 0 as x --> minus infinity (as x decreases) (it often gets
to 0)
F(x) is NONDECREASING (it may be flat or increasing as x increases, but
it's never decreasing)
------- ------ -------
HW:
Some more continuous problems: Handout "Density problems"
Do #1, 3b, 4b, 5
Read 4.2 pp.104 to 114.
Discrete CDF's:
A. Poisson Distribution: Use the
Poisson table handout to find, for lambda = 2.2,
P(X< 4), P(X >
4), P(2 < X < 4), P(2 < X < 4)
B. Use M&M's binomial table,
and construct a table column for B(6, .3), which tables F(x). Also
Graph it.
Find P(2 < X < 4) from M&M's
table, and from your constructed column. Show your work.
(all SPSS built in "tables" are in the CDF
form)
Ash p. 118, #1 Write the formula for F(x)
carefully, paying attention to endpoints. (do
graph both graphs)
Continuous CDF:
Density-->CDF handout: Graph CDF by counting
squares, carefully.
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