document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175333>Question 1175333</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850620 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850620\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 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.\r<BR>\n" );
document.write( "\n" );
document.write( "```python<BR>\n" );
document.write( "import random<BR>\n" );
document.write( "import statistics\r<BR>\n" );
document.write( "\n" );
document.write( "def generate_random_sums(num_samples=50):<BR>\n" );
document.write( "    \"\"\"Generates 50 sums of 5-digit random numbers.\"\"\"<BR>\n" );
document.write( "    sums = []<BR>\n" );
document.write( "    for _ in range(num_samples):<BR>\n" );
document.write( "        random_numbers = [random.randint(10000, 99999) for _ in range(5)]<BR>\n" );
document.write( "        sums.append(sum(random_numbers))<BR>\n" );
document.write( "    return sums\r<BR>\n" );
document.write( "\n" );
document.write( "def calculate_distribution(sums):<BR>\n" );
document.write( "    \"\"\"Calculates the probability distribution, mean, and variance.\"\"\"<BR>\n" );
document.write( "    counts = {}<BR>\n" );
document.write( "    for s in sums:<BR>\n" );
document.write( "        counts[s] = counts.get(s, 0) + 1\r<BR>\n" );
document.write( "\n" );
document.write( "    total_samples = len(sums)<BR>\n" );
document.write( "    probability_distribution = {s: count / total_samples for s, count in counts.items()}\r<BR>\n" );
document.write( "\n" );
document.write( "    mean = statistics.mean(sums)<BR>\n" );
document.write( "    variance = statistics.variance(sums)\r<BR>\n" );
document.write( "\n" );
document.write( "    return probability_distribution, mean, variance\r<BR>\n" );
document.write( "\n" );
document.write( "# Generate the random sums<BR>\n" );
document.write( "random_sums = generate_random_sums()\r<BR>\n" );
document.write( "\n" );
document.write( "# Calculate the distribution, mean, and variance<BR>\n" );
document.write( "distribution, mean, variance = calculate_distribution(random_sums)\r<BR>\n" );
document.write( "\n" );
document.write( "# Print the results<BR>\n" );
document.write( "print(\"Probability Distribution:\")<BR>\n" );
document.write( "for s, prob in sorted(distribution.items()):<BR>\n" );
document.write( "    print(f\"Sum: {s}, Probability: {prob:.4f}\")\r<BR>\n" );
document.write( "\n" );
document.write( "print(f\"\nMean: {mean:.2f}\")<BR>\n" );
document.write( "print(f\"Variance: {variance:.2f}\")<BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "**Explanation and Key Points:**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **`generate_random_sums(num_samples=50)`:**<BR>\n" );
document.write( "    * This function creates a list of 50 sums.<BR>\n" );
document.write( "    * For each sum, it generates 5 random integers between 10000 and 99999 (inclusive).<BR>\n" );
document.write( "    * It then calculates the sum of those 5 random numbers.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **`calculate_distribution(sums)`:**<BR>\n" );
document.write( "    * This function takes the list of sums as input.<BR>\n" );
document.write( "    * It creates a dictionary `counts` to store the frequency of each unique sum.<BR>\n" );
document.write( "    * It calculates the probability of each sum by dividing its frequency by the total number of samples (50).<BR>\n" );
document.write( "    * It uses the statistics library to calculate the mean and variance.\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Output:**<BR>\n" );
document.write( "    * The code prints the probability distribution, showing each unique sum and its corresponding probability.<BR>\n" );
document.write( "    * It then prints the mean and variance of the sums.\r<BR>\n" );
document.write( "\n" );
document.write( "**Important Notes:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Randomness:** Each time you run the code, you'll get different random numbers and therefore a different probability distribution, mean, and variance.<BR>\n" );
document.write( "* **Large Numbers:** The sums will be very large (in the hundreds of thousands), so the probability distribution will likely have many unique values.<BR>\n" );
document.write( "* **Approximation:** With only 50 samples, the probability distribution is an approximation of the true distribution. With more samples, the approximation would become more accurate.<BR>\n" );
document.write( "* **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.<BR>\n" );
document.write( "* **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.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );