Probability Histogram and Central Limit Theorem

[pociteh]

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!

 

[Office_Shredder]

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.

 

[tacman]

Hi there,

I can understand your confusion about the convergence of the empirical histogram to the probability histogram and the normal curve. Let me try to explain it in simpler terms.

Firstly, the empirical histogram is a visual representation of the frequency of occurrence of a particular outcome in a set of data. In your example of tossing a fair coin 25 times, the possible outcomes are either heads or tails, and the empirical histogram will show the number of times each outcome occurred.

Now, as the number of repetitions increases, the empirical histogram will start to resemble the probability histogram. The probability histogram is a theoretical representation of the expected frequency of each outcome in a given set of data. In other words, it shows the probability of each outcome occurring. So, in your example, as the number of repetitions increases, the empirical histogram will start to show equal frequency for both heads and tails, just like the probability histogram.

Moving on to the Central Limit Theorem, it states that as the number of draws (or samples) increases, the probability histogram will start to resemble a normal curve. This means that the distribution of outcomes will be symmetrical around the mean, with most of the outcomes falling close to the mean.

So, to answer your questions:

1. In your first experiment, as the number of repetitions increases, the empirical histogram will start to resemble the probability histogram. However, since the number of draws is only 25, the probability histogram will not resemble a normal curve.

2. In the second experiment, with 100 draws, the probability histogram will start to resemble a normal curve, but the empirical histogram may not fully converge to it.

3. In the third experiment, with 1000 draws, both the empirical histogram and the probability histogram will closely resemble a normal curve.

I hope this helps clarify your doubts. It's important to remember that the convergence to the probability histogram and the normal curve is not instantaneous, but it occurs as the number of repetitions and draws increase. Keep practicing and visualizing these concepts, and it will become clearer over time. Best of luck!