**CppQuant Answer**

- If the return on the asset is -3.52%: z = (-3.52% - 20%)/12% = -1.96.
- If 50.96%: z = (50.96% - 20%)/12% = 2.58.
- For a standard normal random variable X, the 95% confidence interval is -1.96 to 1.96, the 99% confidence interval is -2.58 to 2.58
- A normal distribution is symmetrical, so the probability of z-value falling between -1.96 to 2.58 is 95%/2 + 99%/2 = 97%.

*Bonus Points*

- The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
- There is unlimited number of normal distributions, each with a different mean or standard deviation. Therefore, it's impractical to provide a table of probabilities for each combination of mean and standard deviation. However, we can standardize the actual distribution for a normal random variable to a standard normal distribution with z = (X - m)/s, where X, M, s are a score, the mean, and the standard deviation from the original normal distribution respectively. aka z distribution.
- For a normal random variable X, the exact confidence intervals are
- 90% confidence interval for X is : x-bar - 1.645s to x-bar + 1.645s, with 10% of the observations fall outside the 90% confidence interval, with 5% on each side.
- 95%: x-bar - 1.96s to x-bar + 1.96s: 5% fall outside the 95% confidence interval, with 2.5% on each side.
- 99%: x-bar - 2.58s to x-bar + 2.58s, 1% fall outside the 99% confidence interval, with 0.5% on each side.

*Category: C++ Quant > Probability > Distributions*

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