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When dealing with probability distributions, the expected value (or mean) is a crucial concept. It helps us understand the average outcome we can anticipate over a long period of intervals.

To calculate the expected value, you multiply each possible outcome by its probability. Then, you add up all these products. In our exercise, GESCO Insurance Company can see, on average, what their profit or loss will be from selling a life insurance policy to a 40-year-old female.

In this specific case:

• If the policyholder survives, the company gains \(350 (the premium).
• If she passes away, they incur a loss of \)99,650 (\(100,000 payout minus the \)350 premium).

Mathematically, the expected value \( \mu \) becomes:\[ \mu = (350 \times 0.998) + (-99,650 \times 0.002) = -50 \]
This calculation shows that, on average, the insurance company loses $50 per policy. Understanding this metric allows companies to assess the financial viability of their insurance products.

Standard deviation is a measure of how much variation or dispersion exists from the average (expected) value. In simpler terms, it tells us how spread out the numbers in a data set are.

For GESCO Insurance's policy, this metric helps gauge the potential fluctuation or deviation from the average loss (expected value of $-50).

Here's how to compute it:

• First, you determine the variance, which involves calculating the squared differences from the expected value, weighted by their probabilities.
• Then, take the square root of the variance to get the standard deviation.

The formula looks like this:\[ \sigma^2 = 0.998(350 - (-50))^2 + 0.002(-99,650 - (-50))^2 \]
Finding the standard deviation helps the company understand the risk involved with each policy and adjust their premiums accordingly.

An insurance policy is a contract between the policyholder and the insurance company. It specifies the financial protection against losses in exchange for premium payments.

In our exercise, GESCO Insurance offers a life insurance policy that promises to pay out $100,000 in the unfortunate event of the policyholder's death. In return, the insured person pays a $350 premium per year.

• The premium is the upfront cost policyholders pay to access coverage.
• The payout is the amount the company disburses to beneficiaries if the event (death, in this case) occurs.

This policy provides peace of mind to the insured, knowing their beneficiaries will have financial support if they pass away. For the insurance company, it represents a balance between collecting premiums and potentially making large payouts. Understanding this balance is essential when creating and maintaining profitable insurance products.

In probability and statistics, a random variable is a numerical representation of outcomes from a random phenomenon. It associates possible outcomes of a random event to numerical values.

In the case of GESCO Insurance:

• The random variable \(X\) represents the net gain or loss for the company when a policy is sold.
• Its outcomes are the profit (gain) if the insured survives, or a loss if they pass away.

Two distinct probabilities exist:

• \(X = 350\) with a probability of 0.998 (policyholder surviving).
• \(X = -99,650\) with a probability of 0.002 (policyholder passing away).

Understanding random variables helps in modeling real-world scenarios, such as insurance risk, by assigning numerical probabilities. This enables companies to predict financial performance over time and make more informed decisions.