Question 1175333: Take a 5 digit random numbers of size 50, sum each 5 digit random numbers and consider them as a random variable. Make a probability distribution and find its mean and variance.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
Take a 5 digit random numbers of size 50, sum each 5 digit random numbers and consider them
as a random variable. Make a probability distribution and find its mean and variance.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Mathematically, this post is soup of words with no sense.
First, what is "5 digit random numbers of size 50" ?
Just this one sentence is a wild gibberish, which may to kill a reader to a death.
Next, it says "sum each 5 digit random numbers".
Mathematically, this statement/instruction/sentence is illiterate: its meaning is dark.
What I should add: - the digits of the numbers? - or the numbers themselves ?
Then it should be written "add all 5-digit random numbers", not "sum each 5 digit random numbers."
The authors' writing is extremely illiterate.
The right place of this composition is in the garbage bin.
To the author, I would recommend to take medicine regularly,
to sleep enough and to walk enough, and to forget about mathematical writing.
//////////////////////////////////////////////////
To the managers of this project
Please replace this writer, since he/she is not able to present his/her assignment properly.
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