Let X 1 , X 2 , ... , X n be independent random variables having an unknown continuous distribution function F. and let Y 1 , Y 2 , ... , Y m be independent random variables having an unknown continuous distribution function G. Now order those n + m variables, and let I i = { 1 if the i th smallest of the n + m variables is from the X sample 0 otherwise The random variable R = ∑ i = 1 n + m i I i is the sum of the ranks of the X sample and is the basis of a standard statistical procedure (called theWilcoxon sum-of-ranks test) for testing whether F and G are identical distributions. This test accepts the hypothesis that F = G when R is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of R. Hint: Use the results of Example 3e.
Let X 1 , X 2 , ... , X n be independent random variables having an unknown continuous distribution function F. and let Y 1 , Y 2 , ... , Y m be independent random variables having an unknown continuous distribution function G. Now order those n + m variables, and let I i = { 1 if the i th smallest of the n + m variables is from the X sample 0 otherwise The random variable R = ∑ i = 1 n + m i I i is the sum of the ranks of the X sample and is the basis of a standard statistical procedure (called theWilcoxon sum-of-ranks test) for testing whether F and G are identical distributions. This test accepts the hypothesis that F = G when R is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of R. Hint: Use the results of Example 3e.
Let
X
1
,
X
2
,
...
,
X
n
be independent random variables having an unknown continuous distribution function F. and let
Y
1
,
Y
2
,
...
,
Y
m
be independent random variables having an unknown continuous distribution function G. Now order those
n
+
m
variables, and let
I
i
=
{
1
if the
i
th smallest of the
n
+
m
variables is from the
X
sample
0
otherwise
The random variable
R
=
∑
i
=
1
n
+
m
i
I
i
is the sum of the ranks of the X sample and is the basis of a standard statistical procedure (called theWilcoxon sum-of-ranks test) for testing whether F and G are identical distributions. This test accepts the hypothesis that
F
=
G
when R is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of R.
Hint: Use the results of Example 3e.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Q2
H
let x(+) = &cos (Ait+U) and.
4(+) = ß cos(12t +V), where d. B. 1. In Constants
and U,V indep.rus have uniform dist. (-π,π)
Show that:
①Rxy (+,4+1)=0 @ Rxy (++) = cos [
when U=V
Q3 let x(t) is stochastic process with Wss
-121
e,
and Rx ltst+1) = ( 2, show that
E(X) =
E(XS-X₁)² = 2(-1).
Qu let x(t) = U Cost + (V+1) Sint, tεIR.
where UV indep.rus, and let E (U)-E(V)=0
and E(U) = E(V) = 1, show that
Cov (Xt, Xs) = K (t,s) = cos(s-t) X(+) is not
WSS.
Patterns in Floor Tiling A square floor is to be tiled with square tiles as shown. There are blue tiles on the main diagonals and red tiles everywhere else.
In all cases, both blue and red tiles must be used. and the two diagonals must have a common blue tile at the center of the floor.
If 81 blue tiles will be used, how many red tiles will be needed?
For what numbers in place of 81 would this problem still be solvable?
Find an expression in k giving the number of red tiles required in general.
At a BBQ, you can choose to eat a burger, hotdog or pizza. you can choose to drink water, juice or pop. If you choose your meal at random, what is the probability that you will choose juice and a hot dog? What is the probability that you will not choose a burger and choose either water or pop?
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