Financial Risk Manager Part 1

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A sample of 100 monthly profits gave out the following data:

\sum_{i=1}^{100} x_i = 3,453 \quad \text{and} \quad \sum_{i=1}^{100} x_i^2 = 800,536

What is the sample mean and standard deviation of the monthly profits?

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TTanishq

Sample Mean = 33.53, Standard deviation = 85.99

Sample Mean = 34.53, Standard deviation = 82.96

Sample Mean = 43.53, Standard deviation = 89.99

Sample Mean = 33.63, Standard deviation = 65.99

Explanation:

Calculation Explanation

The sample mean is calculated as:

\hat{\mu} = \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i

\Rightarrow \bar{X} = \frac{1}{100} \times 3,453 = 34.53

The sample variance is calculated using:

s^2 = \frac{1}{n-1} \left\{ \sum_{i=1}^{n} X_i^2 - n \hat{\mu}^2 \right\}

s^2 = \frac{1}{99} (800,536 - 100 \times 34.53^2)

$`$ 34.53`^2 = 1192.3209

$`$100` \times 1192.3209 = 119,232.09

$`$ 800`,536 - 119,232.09 = 681,303.91

s^2 = \frac{681,303.91}{99} = 6881.857677

### Standard Deviation

s = \sqrt{6881.857677} = 82.96

Therefore, the correct values are: - **Sample Mean = 34.53** - **Standard Deviation = 82.96**

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