CPA
Foundation Leval
Quantitative Analysis May 2017
Suggested solutions

2. Infinite Set:

A set that contains an infinite number of elements.

Example: B = {1, 2, 3, ....}

3. Equal Sets:

Two sets that have exactly the same elements.

Example: C = {a, b, c} and D = {c, a, b} are equal sets.

4. Null Set (Empty Set):

A set that contains no elements.

Denoted by ∅ or {}.

Example: E = ∅

5. Singleton Set:

A set that contains only one element.

Example: F = {5}

6. Subset:

If every element of set A is also an element of set B, then A is a subset of B, denoted as A ⊆ B.

Example: If A = {1,2} and B = {1,2,3,4}, then A is a subset of B.

7. Power Set:

The set of all subsets of a given set.

If A = {a,b}, then the power set of A is P(A) = {∅,{a},{b},{a,b}}.

Where:

∅ is the empty set.

{a} is the subset containing only the element 'a'.

{b} is the subset containing only the element 'b'.

{a,b} is the subset containing both 'a' and 'b'.

8. Universal Set:

The set that contains all the elements under consideration in a particular discussion or problem. Denoted by U .

Example: If discussing natural numbers, the universal set might be U = {1, 2, 3, .....}

9. Complement of a Set:

The set of elements not in a given set A with respect to a universal setU. Denoted by A' or A

Example: If U = {1,2,3,4,5} and A = {1,2}, then the complement of A is A′ is = {3,4,5}.

10. Disjoint Sets:

Sets that have no elements in common. Example: If X = {a, b} and Y = {c, d} , then X and Y are disjoint sets.

11. Equal and Equivalent Sets:

Equal sets have the same elements. Equivalent sets have the same cardinality (number of elements).

Example: {1, 2, 3} and {3, 1, 2} are equal sets, while {a, b} and {1, 2} are not equal but equivalent sets.

1. Transition Probability:

In a Markov Chain, a transition probability is the probability of moving from one state to another in a single step. It quantifies the likelihood of transitioning from one state to another in the next iteration of the process. Transition probabilities are often organized into a matrix known as the transition probability matrix.

For a Markov Chain with n states, the transition probability matrix P is an n × n matrix, where P_ij represents the probability of transitioning from state i to state j. The sum of probabilities in each row of the matrix is always 1, as the system must transition to some state.

Transition probabilities play a crucial role in understanding and predicting the behavior of Markov Chains over time.

2. Absorbing State:

An absorbing state in a Markov Chain is a state from which, once entered, the system cannot leave. In other words, an absorbing state is a state in which the system remains indefinitely once it reaches that state. Once the system enters an absorbing state, it will stay in that state with a probability of 1 in all subsequent time steps.

In the context of a transition probability matrix, an absorbing state is characterized by having a diagonal element P_ii = 1 and all other elements in that row and column equal to 0. The absorbing states are often represented by rows and columns where all elements are 0 except for the diagonal element.

The concept of absorbing states is significant in various applications, such as modeling absorbing barriers in stochastic processes, analyzing the long-term behavior of a system, and predicting the ultimate fate of a Markov Chain.

(ii) At maximum profit

(ii) The expenditure incurred by a household whose monthly income is Sh.30,000.

2. Independent Trials:

• The trials are independent, meaning the outcome of one trial does not affect the outcome of any other trial.
• Independence implies that the probability of success or failure remains constant from trial to trial.

3.Constant Probability of Success:

• There are only two possible outcomes for each trial: success and failure.
• The probability of success (denoted as p ) remains constant for each trial.

4. Discrete Random Variable:

• The variable of interest is the number of successes in a fixed number of trials.
• This random variable follows a binomial probability distribution.

(ii) Minimising risk.

(ii) The maximum contribution of C₁ and C₂

(iii) Explaining the effect on contribution of the availability of additional plastic waste and machine time.

An augmentation of one unit in plastic waste will lead to a rise in the total contribution by shs 1733.33. Conversely, a reduction of one unit in plastic waste will cause a decrease in contribution by shs 6800.

For every increase in one unit of machine hours for variable x, there will be a corresponding increase in total contribution by shs 850. Conversely, a reduction of one unit in machine x hours will result in a decrease in total contribution by shs 850.

(iv) The sensitivity of the model to changes in contribution per unit of C₁ and C₂

(v) The increase in contribution of Green Furniture Limited assuming that the management overcomes the plastic waste constraint.

➢ The overall outcome for all players would be zero at the end of the game.

➢ A game is played when each player chooses a course of action (strategy) out of available strategies.

➢ Each outcome determines the gain or loss of each player.

➢ The outcome of the play depends on every combination of the courses of action.

➢ Each player has a finite list of courses of action or strategy.

➢ The number of players is finite.