Distribution & variables

Question 1

Define the Probability Distribution Function (PDF)

Answer 1

P(a

Question 2

Define the Cumulative Distribution Function (CDF)

Answer 2

P(X < x) = f(u) du

Question 3

Define the expected value (mean) of a function

Answer 3

E(X) = x f(x) dx

Question 4

Define the variance of a function

Answer 4

V(X) = E(X^2) - (EX)^2
V(X) = [(x - E(X))^2 f(x) dx](infinity, -infinity)

Question 5

What is the general PDF for a normally distributed function?

Answer 5

f(x) = [1/sqrt(2piV(X))] exp(-(x-E(X))^2/2V(X))

Question 6

The mass of a bag of sand is 50 +/- 0.5kg. What is the mean and variance?

Answer 6

Mean = 50kg
2 Standard deviation = 0.5kg
Variance = (0.025)^2

Question 7

What are the conditions of a standard normal variable?

Answer 7

Mean = 0
Variance = 1

Question 8

How do you standardise a normal independent variable?

Answer 8

Z could be a function of 2 independent variables, eg. 2X + Y.
Z = (x - E(X))/stddev.

Question 9

What tables have the z distribution in?

Answer 9

III

Question 10

Draw the z distribution and label the levels of certainty.

Answer 10

mean +/- 1 stddev. = 68%
mean +/- 2 stddev. = 95%
mean +/- 3 stddev. = 99.7%

Question 11

What is the general PDF for a log-normal distribution function?

Answer 11

f(x) = [1/(xw sqrt(2pi)] exp[-(ln(x) - mean)^2/2w^2]

Question 12

What are the conditions of a log-normal variable?

Answer 12

Y
mean = location = theta
variance = scale = w^2

Question 13

Draw the log-normal distribution and label;
w^2 = 1
w^2 = 0.25
w^2 = 2.25

Answer 13

See diagram

Question 14

Define the mean and variance for;

Y = X + c

Answer 14

Mean = E(Y) = E(X) + c
Variance = V(Y) = V(X) + 0

Question 15

Define the mean and variance for;

Y = c X

Answer 15

Mean = E(Y) = E(cX) = cE(X) 
Variance = V(Y) = V(cX) = c^2 V(X)

Question 16

Considering the volume a cylinder equation. What are the independent variables? Define the mean and variance.

Answer 16

V = pi x r^2 x h
radius and height are both independent.

Mean = E(V) = pi x E(r)^2 x E(h)

Variance = V(V) = [(partial dV/dr)^2 x V(r)] + [(partial dV/dh)^2 x V(h)]