Exercise 4, Page 362

Suppose we choose independently 25 numbers at random (uniform density) from the interval [0, 20]. We are interested in finding the normal densities that approximate the densities of their sum \(S_{25}\), their standardized sum \(S^*_{25}\), and their average \(A_{25}\).

Normal Density Approximations

We are interested in approximating the distributions of the sum \(S_{25}\), the standardized sum \(S^*_{25}\), and the average \(A_{25}\) of 25 independently chosen random numbers from the interval [0, 20] using normal densities.

Sum \(S_{25}\)

The sum \(S_{25}\) is the total of 25 randomly chosen numbers from a uniform distribution between 0 and 20.

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 \(N(250, 28.87)\).

Standardized Sum \(S^*_{25}\)

The standardized sum \(S^*_{25}\) is the sum \(S_{25}\) minus its mean, divided by its standard deviation.

Calculation:

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

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

Average \(A_{25}\)

The average \(A_{25}\) is the mean of 25 uniformly distributed random numbers between 0 and 20.

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.

Sum \(S_{25}\)

The sum \(S_{25}\) is the sum of 25 independent and identically distributed random variables from a uniform distribution over [0, 20].

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:

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

Calculation of Standardized Sum \(S^*_{25}\)

The standardized sum, \(S^*_{25}\), is obtained by subtracting the expected value (mean) and dividing by the standard deviation. The formula is given by:

\[ S^*_{25} = \frac{S_{25} - E[S_{25}]}{\sqrt{\text{Var}(S_{25})}} \]

For \(S_{25}\), which is the sum of 25 independent random variables uniformly distributed from 0 to 20, we have:

Thus, when we standardize \(S_{25}\):

This is because the standardization process shifts the mean to 0 and scales the distribution to have a unit variance.

Average \(A_{25}\)

The average \(A_{25}\) is the sum \(S_{25}\) divided by 25.

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