Theory of Expectation :: Problems on Profits, Business : Probability Distribution

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A company introduces a new product in the market and excepts to make a profit of Rs. 2.5 lakhs during the first year if the demand is "good", a profit of Rs. 1.5 lakhs if the demand is "moderate" and a loss of Rs. 1 lakh if the demand is "poor". Market Research Studies indicate that the probabilities for the demand to be good and moderate are 0.2 and 0.5 respectively.

Find the company's expected profit and the standard deviation of profit.

Net Answers :
[Expectation: +Rs.1.55 lakhs ; Variance: Rs.0.2725 lakhs ; Standard Deviation:+ Rs.0.522 lakhs]

Solution

"x" indicate the profit made by the company during the first year
[Since you are required to find the company's expected profit, the variable would represent the company's profits in the first year]
The profit made by the company during the first year would be

+ Rs. 2.5 laksh if the demand is "good"

+ Rs. 1.5 laksh if the demand is "moderate"

+ Rs. 2.5 laksh if the demand is "poor"

⇒ The values (in lakhs) carried by the variable ("x") would be either 1 or 1.5 or 2.5
⇒ "X" is a discrete random variable with range = {1, 1.5, 2.5}
"X" represents the random variable and P(X = x) represents the probability that the value within the range of the random variable is a specified value of "x"
Probabilty for the demand for the product introduced by the company to be

Good ⇒ P(Good) = 0.2

Moderate ⇒ P(Moderate) = 0.5

Poor ⇒ P(Poor) = 0.3

On the assumption that there are only three possibilities,
i.e. for the demand to be either good, moderate or poor,
The three events "Good", "Moderate" and "Poor" are exhaustive events.
⇒ P(Good ∪ Moderate ∪ Poor) = 1 → (1)
Since only one of the events can occur, the three events "Good", "Moderate" and "Poor" are mutually exclusive
⇒ P(Good ∩ Moderate ∩ Poor) = 0 (Or) P(Good ∪ Moderate ∪ Poor) = P(Good) + P(Moderate) + P(Poor) → (2)
From (1) and (2) we can write
P(Good ∪ Moderate ∪ Poor) = P(Good) + P(Moderate) + P(Poor) = 1
⇒ P(Good) + P(Moderate) + P(Poor) = 1
⇒ 0.2 + 0.5 + P(Poor) = 1
⇒ P(Poor) = 1 − 0.7
⇒ P(Poor) = 0.3
Probability for the profits made by the company (in lakhs) to be

Rs. 2.5 ⇒ P(X = 2.5) = P(2.5)

= 0.2

Rs. 1.5 ⇒ P(X = 1.5) = P(1.5)

= 0.5

+ Rs. 1 ⇒ P(X = + 1) = P(+ 1)

= 0.3

The probabilty distribution of "x" would be

x 1 1.5 2.5

P(X = x) or P 0.3 0.5 0.2

Calculations for Mean and Standard Deviations

x P Px x² Px²

1 0.3 0.3 1 0.3

1.5 0.5 0.75 2.25 1.125

2.5 0.2 0.5 6.25 1.25

Total 1 1.55 2.675

The company's expected earnings
⇒ Expectation of "x"

⇒ E (x) = Σ px

= + Rs. 1.55 lakhs

The company can expect to make a profit of Rs. 1,55,000 in the first year
Variance of the company's profits

⇒ var (x) = E (x²) − (E(x))²

= Σ px² − (Σ px)²

= 2.675 − (1.55)²

= 2.675 − (2.4025)²

= Rs. 0.2725 lakhs

Variance of the companys profits would be Rs. 27,250 approximately
Standard Deviation of the company's profits

⇒ SD (x) = + √ Var (x)

= + √ 0.2725

= + Rs. 0.522 lakhs

Standard Deviation of the companys profits would be Rs. 52,200 approximately

Credit : Vijayalakshmi Desu