Question 1175333
Alright, let's simulate this process and calculate the probability distribution, mean, and variance. Since we're dealing with 5-digit random numbers and summing them, the resulting sums will be quite large, and it's best handled with a programming language like Python.

```python
import random
import statistics

def generate_random_sums(num_samples=50):
    """Generates 50 sums of 5-digit random numbers."""
    sums = []
    for _ in range(num_samples):
        random_numbers = [random.randint(10000, 99999) for _ in range(5)]
        sums.append(sum(random_numbers))
    return sums

def calculate_distribution(sums):
    """Calculates the probability distribution, mean, and variance."""
    counts = {}
    for s in sums:
        counts[s] = counts.get(s, 0) + 1

    total_samples = len(sums)
    probability_distribution = {s: count / total_samples for s, count in counts.items()}

    mean = statistics.mean(sums)
    variance = statistics.variance(sums)

    return probability_distribution, mean, variance

# Generate the random sums
random_sums = generate_random_sums()

# Calculate the distribution, mean, and variance
distribution, mean, variance = calculate_distribution(random_sums)

# Print the results
print("Probability Distribution:")
for s, prob in sorted(distribution.items()):
    print(f"Sum: {s}, Probability: {prob:.4f}")

print(f"\nMean: {mean:.2f}")
print(f"Variance: {variance:.2f}")
```

**Explanation and Key Points:**

1.  **`generate_random_sums(num_samples=50)`:**
    * This function creates a list of 50 sums.
    * For each sum, it generates 5 random integers between 10000 and 99999 (inclusive).
    * It then calculates the sum of those 5 random numbers.

2.  **`calculate_distribution(sums)`:**
    * This function takes the list of sums as input.
    * It creates a dictionary `counts` to store the frequency of each unique sum.
    * It calculates the probability of each sum by dividing its frequency by the total number of samples (50).
    * It uses the statistics library to calculate the mean and variance.

3.  **Output:**
    * The code prints the probability distribution, showing each unique sum and its corresponding probability.
    * It then prints the mean and variance of the sums.

**Important Notes:**

* **Randomness:** Each time you run the code, you'll get different random numbers and therefore a different probability distribution, mean, and variance.
* **Large Numbers:** The sums will be very large (in the hundreds of thousands), so the probability distribution will likely have many unique values.
* **Approximation:** With only 50 samples, the probability distribution is an approximation of the true distribution. With more samples, the approximation would become more accurate.
* **Theoretical Distribution:** The theoretical distribution of the sums would approach a normal distribution due to the central limit theorem, as you are summing multiple independent random variables.
* **Computational power:** If you wanted to do this by hand, it would take an extremely long time. That is why code is the best way to handle this problem.