To find the variance of the MRA Company's profits, we use the following steps:

Step 1: Formula for Variance

The variance $\sigma^2$ of a probability distribution is calculated by the formula:

$\sigma^2 = \sum [P(x) \times (x - \mu)^2]$

Where:

• $x$ is the value of the random variable (profit in $1,000s),
• $P(x)$ is the probability associated with that value,
• $\mu$ is the expected value (mean) of the distribution, which is calculated as:

$\mu = \sum [P(x) \times x]$

Step 2: Calculate the Expected Value $\mu$

We first need to compute the expected value $\mu$.

Given dataTo find the variance of the MRA Company's profits, we use the following steps:

Step 1: Formula for Variance

The variance $\sigma^2$ of a probability distribution is calculated by the formula:

$\sigma^2 = \sum [P(x) \times (x - \mu)^2]$

Where:

• $x$ is the value of the random variable (profit in $1,000s),
• $P(x)$ is the probability associated with that value,
• $\mu$ is the expected value (mean) of the distribution, which is calculated as:

$\mu = \sum [P(x) \times x]$

Step 2: Calculate the Expected Value $\mu$

We first need to compute the expected value $\mu$.

Given

x &: -200, -75, 0, 50, 125, 250, 375 \\ P(x) &: 0.07, 0.09, 0.21, 0.31, 0.24, 0.06, 0.02 \\ \end{aligned}$$ Now calculate $$\mu$$: $$\mu = (-200 \times 0.07) + (-75 \times 0.09) + (0 \times 0.21) + (50 \times 0.31) + (125 \times 0.24) + (250 \times 0.06) + (375 \times 0.02)$$ Let me calculate this for you.The expected value $$\mu$$ is 47.25 (in $1,000s). ### Step 3: Calculate the Variance $$\sigma^2$$ Now, we can compute the variance using the formula: $$\sigma^2 = \sum [P(x) \times (x - \mu)^2]$$ I'll calculate this for you next.The variance in profit for the MRA Company for the first year of operation is $$12161.1875$$ (in $1,000²). Would you like further details or have any questions? Here are 5 related questions you could explore: 1. How do you interpret the meaning of variance in this context? 2. What is the standard deviation of the profits based on this variance? 3. How would an increase in the probabilities of extreme profits affect the variance? 4. What is the probability that the profit will be more than $$100$$ in the first year? 5. Can you compute the skewness of this profit distribution? **Tip:** Variance measures how spread out the profits (or losses) are from the expected value, giving insight into the risk involved in the project.