Reference no: EM131037048

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.]