Struggling with lecture regarding Central Limit Theorem - Continuous Random Variable

How are these even calculated. This lecture is poorly explained.

At 1:20 the instructor says let’s start with n equals 1 where you’re just averaging one weight time.
Let’s repeat this experiment of averaging one weight time many, many,
many times to get a nice histogram and see the distribution.
So the histogram looks like this.
It kind of looks like a uniform density from far away because
each sample of y1 comes from a uniform 0 to 15 distribution.
Of course, there are ups and downs.

How is this even calculated? and what does density means in this context (y-axis of histogram)?

Hey there @Musab_Bin_Gulfam

In the image, a uniform distribution between 0 and 15 is used, meaning every possible value in this range is equally likely. When n = 1, you’re taking a single sample from this uniform distribution. Mathematically, if you draw a sample X_1 from a uniform distribution U(0, 15), then Y_1 = \frac{X_1}{1} = X_1.

Density on the y-axis represents the relative frequency of the sampled values. It shows how often values occur in different intervals. For a uniform distribution, you expect each value between 0 and 15 to occur with approximately equal frequency, so the density should be roughly constant across the entire range.

Since each sample Y_1 is directly taken from the uniform distribution [0, 15], the resulting histogram shows a uniform distribution of values. As you increase the number of samples, the histogram will more accurately builds the true uniform distribution.

According to the Central Limit Theorem, as you increase n (the number of samples you average), the distribution of the average of these samples will approach a normal distribution, even if the original distribution is not normal. When n is large, the histogram of these averages will start to resemble a bell curve instead of the uniform distribution.

Hope this helps, feel free to ask if you need further assistance!

1 Like