Probability Histogram and Central Limit Theorem

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SUMMARY

The discussion focuses on the convergence of empirical histograms to probability histograms and their relationship to the Central Limit Theorem (CLT). It is established that as the number of repetitions increases, the empirical histogram converges to the probability histogram. Furthermore, as the number of draws increases, the probability histogram approaches a normal curve, confirming the principles of the CLT. The conversation clarifies that tossing a coin multiple times constitutes a single experiment, and repeating this experiment numerous times yields a distribution that approximates a normal distribution.

PREREQUISITES
  • Understanding of empirical histograms and probability histograms
  • Familiarity with the Central Limit Theorem (CLT)
  • Basic knowledge of statistical experiments and data collection
  • Concept of discrete data and its graphical representation
NEXT STEPS
  • Study the implications of the Central Limit Theorem in various statistical contexts
  • Explore the concept of empirical distributions and their applications
  • Learn about the differences between discrete and continuous probability distributions
  • Investigate methods for plotting and interpreting histograms in statistical analysis
USEFUL FOR

Students, statisticians, and data analysts seeking to deepen their understanding of statistical convergence, empirical data analysis, and the Central Limit Theorem.

pociteh
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Hi,


I have trouble understanding the convergence of empirical histogram to probability histogram and the convergence of empirical histogram to normal curve.

It was written in my lecture notes that as the number of repetitions goes large, empirical histogram converges to probability histogram, and as the number of draws goes large, probability histogram converges to the normal curve (Central Limit Theorem). It was also said that if the number of repetitions and number of draws are both large, the empirical histogram converges to the normal curve.

Sounds OK so far, but I still have doubts:

1. Suppose I toss a fair coin 25 times and count the number of heads. As the number of repetition goes large, does the empirical histogram converge to probability histogram and then the probability histogram converge to normal curve, or does the empirical histogram only converge to probability histogram? Also, here, the number of draws = 25 and the number of repetitions is x (x keeps increasing), right? (I still kind of confuse the term 'draws' and 'repetitions' at times)

2. Suppose I do another experiment similar to no (1), but I toss it 100 times. Same question.

3. Suppose I do the same experiment again, but I toss 1000 times. Same question.



Please help enlighten. Thank you!
 
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So what would happen is tossing a coin 25 times is a single experiment. You might then repeat that experiment 10,000 times, plotting each time what the number of heads is. As you do this more often, your plot will start to look like a normal distribution centered around 12.5. It'll be rough because you have discrete data points of course, so you have to extrapolate what the curve should look like between them.

For the 100 tosses, again you would measure the number of heads in a 100 toss experiment. You expect around 50. Then if you perform the experiment a large number of times (say 10,000), you get a bunch of data points for the number of heads ranging from 0 to 100, and as you plot more points, it resembles the normal distribution.

Etc.
 

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