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.
Let X1 and X2 be discrete random variables with joint probability distribution
x1 = 1, 2; x2 = 1, 2, 3,
f(x1, x2) =
10,
18
elsewhere.
Find the probability distribution of the random variable Y = X1X2.
Determine ?(?>2).
Let X1, X2, and.X3 be independent and normally distributed random variables with E(X1)
4, E(X2) = 3, E(X3) = 2, Var(X1) = 1, Var(X2) = 5, Var(X3) = 2. Let Y = 2X1 + X2 – 3X3. Find
2.
the distribution of Y.
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