Questions-
1. The probability distributions of the random variables X, Y = X + 6 and Z = X2 - 2 are
|
Values of X
|
-18
|
0
|
3
|
4
|
|
Probability
|
4/37
|
3/37
|
18/37
|
12/37
|
|
Values of Y
|
-12
|
6
|
9
|
10
|
|
Probability
|
4/37
|
3/37
|
18/37
|
12/37
|
|
Values of Z
|
322
|
-2
|
7
|
14
|
|
Probability
|
4/37
|
3/37
|
18/37
|
12/37
|
a) Find E[X] , E[Y] and E[Z] .
b) Check your answers for E[Y] using the linear function rule.
c) Does this rule work in the case of E[Z]?
d) Find E[X2], E[Y2] and E[Z2] and then var [X] , var [Y] and var [Z] .
e) Check the relationship of var [X] and var [Y] through the linear function rule for variances.
f) Does this seem to apply to var [Z] ?
2. This question uses a random variable V with probability density function
1/6 for -3 < v ≤ 3
f (v) =
0 otherwise
a) Using a sketch of the probability density function for the random variable V, find E[V] by a purely geometric argument involving areas.
b) Confirm your answer by finding E[V] using an integration argument.
c) [Harder] Find E[V2] by an integration argument.
d) Use the value found in parts (a) and (b) for E [V] and the value for E[V2] found in part (c) to find var [V].
e) Define new random variables
W = V + 2, Z = 2V + 1.
What are the ranges of values that W and Z can take on?
f) Find the cumulative probability distributions of W and Z.
g) Find
i. Pr (W ≤ 0); Pr (W ≤ 1); Pr (W ≤ 2);
ii. Pr (Z ≤ -2); Pr (Z ≤ 1); Pr (Z ≤ 4); Pr (Z ≤ 6).
3. This question creates a discrete random variable from a continuous random variable.
The tensile quality of a steel billet can be measured by its α- content. The α- content is known to be normally distributed with mean 2 and variance 9. When the α- content is less than or equal to -2, the price charged per tonne of steel is £250. When the α- content is between -2 and 0, the price is £400. When the α- content is between 0 and 6, the price is £500, and when the α- content exceeds 6, the price is £1000 per tonne.
Define a suitable discrete random variable to represent the price of a tonne of steel. What are the values of this random variable? What are the corresponding probabilities? Find the probability distribution and expected value of this discrete random variable. [Hint: you will need to do a number of Normal probability calculations to find the required probability distribution and use the Red Book of Statistical tables.]