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.
Q1. A group of five applicants for a pair of identical jobs consists of three men and two
women. The employer is to select two of the five applicants for the jobs. Let S
denote the set of all possible outcomes for the employer's selection. Let A denote
the subset of outcomes corresponding to the selection of two men and B the subset
corresponding to the selection of at least one woman. List the outcomes in A, B,
AUB, AN B, and An B. (Denote the different men and women by M₁, M2, M3
and W₁, W2, respectively.)
Q3 (8 points)
Q3. A survey classified a large number of adults according to whether they were diag-
nosed as needing eyeglasses to correct their reading vision and whether they use
eyeglasses when reading. The proportions falling into the four resulting categories
are given in the following table:
Use Eyeglasses for Reading
Needs glasses Yes
No
Yes
0.44
0.14
No
0.02
0.40
If a single adult is selected from the large group, find the probabilities of the events
defined below. The adult
(a) needs glasses.
(b) needs glasses but does not use them.
(c) uses glasses whether the glasses are needed or not.
4. (i) Let a discrete sample space be given by
N = {W1, W2, W3, W4},
and let a probability measure P on be given by
P(w1) = 0.2, P(w2) = 0.2, P(w3) = 0.5, P(wa) = 0.1.
Consider the random variables X1, X2 → R defined by
X₁(w1) = 1, X₁(w2) = 2,
X2(w1) = 2, X2 (w2) = 2,
Find the joint distribution of X1, X2.
(ii)
X1(W3) = 1, X₁(w4) = 1,
X2(W3) = 1, X2(w4) = 2.
[4 Marks]
Let Y, Z be random variables on a probability space (, F, P).
Let the random vector (Y, Z) take on values in the set [0, 1] x [0,2] and let the
joint distribution of Y, Z on [0, 1] x [0,2] be given by
1
dPy,z (y, z) ==(y²z+yz2) dy dz.
harks 12 Find the distribution Py of the random variable Y.
[8 Marks]
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