Probability And Statistics

Question 1

Find the probability of a joint distribution function

Answer 1

(1) draw the boundaries using the interval (box)
(2) rearrange the probability and draw this on the graph and find the area of the region.
(3) if the area is found simply then multiply the ans by the joint distribution value.
Otherwise construct a double integral for the area and this is the answer.

Question 2

For a joint distribution function what is the expectation and how do you show independence?

Answer 2

E(x)=Integral x fX(x) dx
E(y)=Integral y fY(y) dx
Independence is fX,Y(x,y)=fX(x)fY(y)

Question 3

What are the steps for change of variables?

Answer 3

(1) rearrange the new variables to make x and y the subject
(2) construct the Jacobean matrix and find the determinant, (dx/dudy/dv-dx/dvdy/du) and take the modulus of this value.
(3) multiply this matrix by the joint distribution function where x and y have been replaced in terms of u and v.
(4) and simplify to obtain the joint density.

Question 4

What is the formula for the conditional probability?
And the expectation for x|y
What is the equation for the COV(X,Y)?

Answer 4

For fX|Y(x|y)=fX,Y(x,y)/fY(y)
And visa versa for y|x
For the expectation: E(X|Y)=sum of xfX|Y(x|y)
Cov(X,Y)=E(XY)-E(X)E(Y) where
E(XY)=sum of xy*fX,Y(x,y) if the covariance=0 then the marginals are independent.

Question 5

Write down all of the summary statistics.

And how you can check whether the model is fit for linear regression analysis.

Answer 5

Check in book.
If the correlation |Sxx/sq(Sxx*Syy)|>0.8
Write the linear regression equation:
Y=ax+b where a=Sxy/Sxx and 
b=(y-bar) - (x-bar)*a
Another check is that the error variance is small compared to the y values. 
1/n(Syy-Sxy^2/Sxx)
Also residuals should be between +- 2 
standard deviations where residuals: 
e=y - y' where y is the original y values subtract the linear regression equation y values.

Question 6

Write the equation for the form of the exponential family of distributions.
How do you find the normal form?

Answer 6

Check in book.
Normal form is obtained by parameterising the eta term by rearranging and making the other term the subject. Substitute this into the original form and sort into the other EFD form. Where the A(eta) can be used to find the expectation and variance, first and second derivative respectively.

Question 7

How do you find the MLE?

Answer 7

(1) write L=product from i=1 to n of the function
(2) then write l= sum of the log of the function
(3) then apply the log function to the function
(4) where x is involved replace with the sum from i=1 to n of xi.
(5) then differentiate with respect to the parameter we want and set to 0.
(6) 1/n*sum of xi is the x-bar.
(7) make the parameter-bar the subject.

Question 8

How do you find an unbiased estimator for the parameter?

Answer 8

Want to show that the expectation of the parameter-bar is equal to the parameter.

Question 9

How do you find the moment-generating function?

Answer 9

Mx(t)=E(exp(tx))=exp(tx)*function
For the kth moment we have that
E(X^k)=d^k/dt^k Mx(t) at t=0
So the expectation is the first derivative at t=0.

Question 10

How to find C for the joint distribution function?

And the x and y marginals?

Answer 10

Do the double integral.

Check in book

Question 11

What is the formula for the Z-Test and

T-Test.

Answer 11

Check in book.
For t-test is when the variance is unknown.

For a sample variance s^2, it is calculated by 1/n-1*sum of (xi-(x-bar))^2.

Question 12

For a difference in means of populations, what are the three cases and equations that follow?

Answer 12

(1) when the variance is unknown but are known to be equal we use the pooled t-test ~ t(m+n-2). Write equations and check in book.
(2) when the population variances are not the same by the sample sizes are sufficiently large for it to work we use the z-test but with the unknown variances Sx and Sy.
(3) the two variances are known and are equal we use the z-test.

Question 13

What are the steps to test the hypothesis for a contingency table?

Answer 13

(1) state the hypothesis
(2) calculate the proportion p=x/n
(3) compute the expected frequencies p*n total =fe
(4) draw the table, so we obtain to total for (f0-fe)^2/fe
(5) degrees of freedom: (n-1)(m-1) n=rows
(6) we accept for

Question 14

What is the test for the value of a variance of a normal population?

Answer 14

V=(n-1)S^2/variance0 in the hypothesis~chi squared (n-1)

If |V|

Question 15

How to test for the ratio of variances of two normal populations.

Answer 15

We use the F-test:
F=SA^2/SB^2 for SA^2>SB^2

For the numerator the d.f. N-1
And the denominator d.f M-1
So we find F(1-alpha/2, n-1, m-2)

Question 16

For a proportion test what do you do? How are p-values calculated?

Answer 16

Z=p-p0/(square root (p0(1-p0)/n)) where p is the proportion and p0 is the p stated in the hypothesis.
Then to test whether we reject or not follow as if this was the z test.

For a two tailed test:
P value: 2P(Z>z)
Left tailed:
P(Zz)
For a p-value