Financial Risk Manager Part 1

Explanation

Given:

• X ~ N(3000, 400²) - Normal distribution with mean μ = 3000 and standard deviation σ = 400
• We need to find P(X > 3500)

Step 1: Standardize the variable [Z = \frac{X - \mu}{\sigma} = \frac{3500 - 3000}{400} = \frac{500}{400} = 1.25]

Step 2: Find the probability [P(X > 3500) = P(Z > 1.25)]

Step 3: Use the standard normal table From the standard normal table:

• P(Z < 1.25) = 0.8944
• Therefore: P(Z > 1.25) = 1 - P(Z < 1.25) = 1 - 0.8944 = 0.1056

Verification:

• The standard normal distribution is symmetric
• About 68% of values lie within ±1 standard deviation
• About 95% of values lie within ±2 standard deviations
• Since 3500 is 1.25 standard deviations above the mean, the probability of exceeding this value should be relatively small, which matches our calculated result of 0.1056 (about 10.56%).