Question 1191307: Company ABC estimates the net profit on a new product it is launching to be Rs. 3,000,000 if it is successful ; Rs. 1,000,000 if it is 'moderately successful' and a loss of Rs. 1,000,000 if it is 'unsuccessful'. The firm assigns the following probabilities to the different possibilities :Successful 0-15, 'moderately successful' 0-25 and unsuccessful 0-60. Find the expected value and variance of the net profit.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the expected value and variance of the net profit:
**1. Define the Random Variable:**
Let X be the random variable representing the net profit.
**2. Create the Probability Distribution:**
| Outcome | Net Profit (X) | Probability (P(X)) |
|---|---|---|
| Successful | Rs. 3,000,000 | 0.15 |
| Moderately Successful | Rs. 1,000,000 | 0.25 |
| Unsuccessful | -Rs. 1,000,000 | 0.60 |
**3. Calculate the Expected Value (Mean):**
The expected value E(X) is calculated as the sum of each possible value of X multiplied by its probability:
E(X) = (3,000,000 * 0.15) + (1,000,000 * 0.25) + (-1,000,000 * 0.60)
E(X) = 450,000 + 250,000 - 600,000
E(X) = 100,000
Therefore, the expected net profit is Rs. 100,000.
**4. Calculate the Variance:**
The variance Var(X) is calculated as the expected value of the squared difference between each value of X and the mean:
Var(X) = Σ[(X - E(X))² * P(X)]
Var(X) = (3,000,000 - 100,000)² * 0.15 + (1,000,000 - 100,000)² * 0.25 + (-1,000,000 - 100,000)² * 0.60
Var(X) = (2,900,000)² * 0.15 + (900,000)² * 0.25 + (-1,100,000)² * 0.60
Var(X) = 8,410,000,000,000 * 0.15 + 810,000,000,000 * 0.25 + 1,210,000,000,000 * 0.60
Var(X) = 1,261,500,000,000 + 202,500,000,000 + 726,000,000,000
Var(X) = 2,190,000,000,000
Therefore, the variance of the net profit is Rs. 2,190,000,000,000.
**5. Calculate the Standard Deviation:**
The standard deviation SD(X) is the square root of the variance:
SD(X) = √Var(X)
SD(X) = √2,190,000,000,000
SD(X) ≈ Rs. 1,479,864.86
**Summary of Results:**
* **Expected Value (Mean):** Rs. 100,000
* **Variance:** Rs. 2,190,000,000,000
* **Standard Deviation:** Rs. 1,479,864.86
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