Let X denote the profit of a business man. The probability of getting profit `Rs. 3000` is 0.6. The probability of getting loss `Rs 4000 ` is 0.3. The probability of getting neither profit nor loss is 0.1 . The mean and variance of X are

Options

A. 100, 182000000
B. 4,00, 4560000
C. 400, 12300
D. 600, 984 0000

Answer

To solve the problem, we need to find the mean and variance of the random variable \( X \), which represents the profit of a businessman. The given probabilities and outcomes are as follows:

- Profit of Rs. 3000 with a probability of 0.6
- Loss of Rs. 4000 with a probability of 0.3 (we will represent this as -4000)
- Neither profit nor loss (i.e., Rs. 0) with a probability of 0.1

### Step 1: Define the values of \( X \) and their probabilities

We can summarize the information in a table:

| Outcome (X) | Probability (P(X)) |
|-------------|---------------------|
| 3000 | 0.6 |
| -4000 | 0.3 |
| 0 | 0.1 |

### Step 2: Calculate the Mean (\( \mu \))

The mean of a random variable is calculated using the formula:

\[ \mu = E(X) = \sum (X \cdot P(X)) \]

Calculating each term:

1. For \( X = 3000 \):
\[ 3000 \cdot 0.6 = 1800 \]

2. For \( X = -4000 \):
\[ -4000 \cdot 0.3 = -1200 \]

3. For \( X = 0 \):
\[ 0 \cdot 0.1 = 0 \]

Now, summing these values:

\[ \mu = 1800 - 1200 + 0 = 600 \]

### Step 3: Calculate the Variance (\( \sigma^2 \))

The variance of a random variable is calculated using the formula:

\[ \sigma^2 = E(X^2) - (E(X))^2 \]

First, we need to calculate \( E(X^2) \):

1. For \( X = 3000 \):
\[ 3000^2 \cdot 0.6 = 9000000 \cdot 0.6 = 5400000 \]

2. For \( X = -4000 \):
\[ (-4000)^2 \cdot 0.3 = 16000000 \cdot 0.3 = 4800000 \]

3. For \( X = 0 \):
\[ 0^2 \cdot 0.1 = 0 \]

Now, summing these values:

\[ E(X^2) = 5400000 + 4800000 + 0 = 10200000 \]

Now we can calculate the variance:

\[ \sigma^2 = E(X^2) - (E(X))^2 = 10200000 - (600)^2 \]
\[ \sigma^2 = 10200000 - 360000 = 9876000 \]

### Final Results

Thus, the mean and variance of \( X \) are:

- Mean (\( \mu \)) = 600
- Variance (\( \sigma^2 \)) = 9876000

### Summary

The final answer is:

\[ \text{Mean} = 600, \quad \text{Variance} = 9876000 \]