Calculation:

• Mean ($\mu_{S_{25}}$): The mean of a single uniform random number between 0 and 20 is $\frac{0 + 20}{2} = 10$. For 25 such independent numbers, $\mu_{S_{25}} = 25 \times 10 = 250$.
• Variance ($\sigma^2_{S_{25}}$): The variance for a single uniform number between 0 and 20 is $\frac{(20 - 0)^2}{12}$. Thus, for 25 numbers, $\sigma^2_{S_{25}} = 25 \times \frac{400}{12} \approx 833.33$.
• Standard Deviation ($\sigma_{S_{25}}$): $\sqrt{833.33} \approx 28.87$.

Therefore, the normal density approximating $S_{25}$ is $N(250, 28.87)$.

Calculation:

• Mean ($\mu_{S^*_{25}}$): 0
• Standard Deviation ($\sigma_{S^*_{25}}$): 1

Hence, the normal density that approximates $S^*_{25}$ is $N(0, 1)$.

Calculation:

• Mean ($\mu_{A_{25}}$): The mean is the same as that of a single number, $10$.
• Variance ($\sigma^2_{A_{25}}$): $\frac{400}{12 \times 25} \approx 1.3333$.
• Standard Deviation ($\sigma_{A_{25}}$): $\sqrt{1.3333} \approx 1.15$.

The normal density that approximates $A_{25}$ is $N(10, 1.15)$.

By using the Central Limit Theorem, these normal densities provide a good approximation for the distributions of the sum, standardized sum, and average of the 25 randomly chosen numbers.

Calculations

• The expected value (mean) of a single uniform random variable $U(0, 20)$ is given by:

$$E[U(0, 20)] = \frac{0 + 20}{2} = 10$$

• For the sum of 25 such random variables:

$$E[S_{25}] = 25 \times E[U(0, 20)] = 25 \times 10 = 250$$

• The variance of a single uniform random variable is:

$$\text{Var}(U(0, 20)) = \frac{(20 - 0)^2}{12} = \frac{400}{12} \approx 33.33$$

• For the sum of 25 such random variables:

$$\text{Var}(S_{25}) = 25 \times \text{Var}(U(0, 20)) = 25 \times 33.33 \approx 833.33$$

• Therefore, the normal density approximating $S_{25}$ is:

$$S_{25} \sim N(250, \sqrt{833.33})$$

Calculations

• The expected value is:

$$E[A_{25}] = \frac{E[S_{25}]}{25} = \frac{250}{25} = 10$$

• The variance is:

$$\text{Var}(A_{25}) = \frac{\text{Var}(S_{25})}{25^2} = \frac{833.33}{625} \approx 1.33$$

• So, the normal density approximating $A_{25}$ is:

$$A_{25} \sim N(10, \sqrt{1.33})$$

Below is the r code to deduce these normal approximations for $S_{25}$, $S^*_{25}$, and $A_{25}$.

# Set seed for reproducibility
set.seed(123)

# Generate 25 random numbers from U(0, 20)
random_numbers <- runif(25, min = 0, max = 20)

# Sum S_25
S_25 <- sum(random_numbers)

# Standardized Sum S*_25
mean_S_25 <- mean(random_numbers) * 25
std_dev_S_25 <- sqrt(25 * (20^2 / 12))
S_star_25 <- (S_25 - mean_S_25) / std_dev_S_25

# Average A_25
A_25 <- mean(random_numbers)

# Display results
cat("Sum S_25:", S_25, "\n")

## Sum S_25: 297.7799

cat("Standardized Sum S*_25:", S_star_25, "\n")

## Standardized Sum S*_25: 0

cat("Average A_25:", A_25, "\n")

## Average A_25: 11.91119