Exercise
Answer
We are given information about a series of mutually independent and identically distributed contributions.
| Parameter | Value |
|---|---|
| Number of contributions ([math]n[/math]) | 2025 |
| Mean of a single contribution ([math]\mu[/math]) | 3125 |
| Standard deviation of a single contribution ([math]\sigma[/math]) | 250 |
Let [math]S[/math] be the total contributions received. Since the contributions are mutually independent and identically distributed, the mean of the total contributions is the sum of the individual means, and the variance of the total contributions is the sum of the individual variances.
- Mean of Total Contributions ([math]E[S][/math]):
The mean of the sum of [math]n[/math] independent and identically distributed random variables is [math]n[/math] times the mean of a single variable.
- Standard Deviation of Total Contributions ([math]SD[S][/math]):
The variance of the sum of [math]n[/math] independent and identically distributed random variables is [math]n[/math] times the variance of a single variable. The standard deviation is the square root of the variance.
According to the Central Limit Theorem, for a large number of independent and identically distributed random variables, the distribution of their sum (or average) approaches a normal distribution, regardless of the shape of the individual distributions. Given that there are 2025 contributions, which is a large number, the total contributions [math]S[/math] can be approximated by a normal distribution with the mean and standard deviation calculated in Step 2.
To find the 90th percentile of a normally distributed variable, we need to find the Z-score that corresponds to a cumulative probability of 0.90. From standard normal distribution tables or calculators, the Z-score for the 90th percentile ([math]Z_{0.90}[/math]) is approximately 1.282.
The formula for calculating a specific percentile of a normal distribution is:
- Mean of total contributions ([math]E[S][/math]): 6,328,125
- Standard deviation of total contributions ([math]SD[S][/math]): 11,250
- Z-score for 90th percentile ([math]Z_{0.90}[/math]): 1.282
- The Central Limit Theorem (CLT) is crucial for approximating the distribution of a sum (or average) of many independent and identically distributed random variables with a normal distribution.
- For a sum of [math]n[/math] independent and identically distributed random variables, if each has mean [math]\mu[/math] and standard deviation [math]\sigma[/math], the sum will have a mean of [math]n\mu[/math] and a standard deviation of [math]\sqrt{n}\sigma[/math].
- Percentiles of a normal distribution are calculated using the formula [math]\text{Percentile} = \text{Mean} + Z \times \text{Standard Deviation}[/math], where [math]Z[/math] is the Z-score corresponding to the desired percentile.