3-4-9. Normal Distribution Theory

The next thing I’ll do is I’ll divide it by σ². Without telling you why I’m doing this, I want to see what the effect is. Suppose σ² = 4. That means we have a variance of 4 and a standard deviation of 2. I’ve given you already the quadratic function when it isn’t divided … Read more

I would submit that it’s this one over here. The reason is this expression is 0 when x = µ. As a result, it can’t be the fourth of these choices. It’s strictly non-negative, so it can’t go down into the negative area, so this one is being out ruled. It’s quadratic. Hence, a function … Read more

I will define for you a normal distribution with a specific mean that’s often called µ, Greek letter µ, and a variance that’s often called σ². We already know that variance is a quadratic expression. In normal land we often use µ and σ². Let’s do this. The very first element is that for any … Read more

The answer is a resounding yes. Even more so, what I’m going to show you doesn’t just apply to binomial distributions with fair coins. It applies to almost any distribution that is sampled many, many times, which is a very deep statistical result. I will construct for you the formula that is being used.

The question really is can we find a better formula for this bell-shaped curve. The answer is, well, take your guess.

And it’s this one. Let me show you. Here’s a typical one, and I apologize the axis over here can’t really be read, but you can take with faith that the center is 0.5, and you can see the characteristic bell curve for this simple coin-flipping experiment. For this run the mean was 0.50006. If … Read more

The other interesting thing is things fall down in the interesting fashion as you deviate from 10, 11, 12, 13, 14 all the way to 0 or 20. Obviously we got a curve that looks a bit like this. This curves is often called a bell curve because it’s quite feasible to think of it … Read more

What I’ve shown you in the beginning of class have from a coin flip to a binomial distribution all the way to a normal distribution, and you might think that this was challenging and indeed it was. As it turns out, you can treat all this things about the same. In fact, if you’re a … Read more

As I’m sure you’ve guessed this is the probability for a single coin flip. This is the formula we can use for multiple coin flips and as we go to very large numbers, the [UNKNOWN] is often a good approximation for the outcome of many coin flips. I should warn you, if the coin flip … Read more

So let’s look for a second at different ways to compute probabilities for coin flips. You have a coin that has probability P of coming up heads. I will give you now 3 very different formulas. Here’s a single probability, here’s our common [UNKNOWN] formula and here’s our Gaussian exponential. So what I want to … Read more

So, here is our normal distribution again. I’m going to write it as “exp” for exponential {-½ (x – µ)²/σ²}. The truth is when you’re new to this this looks really cryptic. When you’re with statistics for many years as I have been, you wake up in the middle of the night and you can … Read more

We have a function f that assumes the value 1 when x = µ that goes to 0 when x goes to ±∞. It so happens that it looks like a bell curve. The fact that it looks like a bell curve is not entirely obvious, but you have to take my word for it. … Read more

The answer is 0. As x goes to infinity, this expression goes to negative infinity. The exponential of negative infinity converges to 0. So, now we’ve got basically the normal distribution function.

In fact, what do you think is the value of this where x = ∞?

The answer is now ± ∞. If you look at this, if you put a really large positive or negative value in, the difference to any µ will be enormous. The square will be even more enormous. Therefore, this entire expression on the right side will be huge. Put a minus sign in front of … Read more

I just programmed and ran the experiment, and the answer is 10. The reason why the answer is 10 is because the number of combinations to place 10 positives and 10 negatives into our list of 20 is larger than any other number. This term over here is maximized when k is exactly half of … Read more

Next I’d like to know where is f(x) minimized? For what value of x would we get the possible smallest value of this entire expression over here? Again, pick one or more of those choices over here.

Quite interestingly, even though this formula looks complex, it a really easy answer, which is when x = µ this thing here is 0. That makes the entire thing 0. e⁰–any value to the 0–is going to be 1.

Let me ask you another question. What is the value of this function if we go to the point where it’s maximum, which is x = µ? That’s the way to write this. Compute for me in your head this what this thing will be when x = µ.

to understand the solution, it’s useful to draw the exponential function. e⁰ is 1 and then it goes up exponentially to really large numbers. If you’ve ever heard Ray Kurzweil talk about the future of society, you’ve seen these curves–everything goes up exponentially. Everything is just exponential. And further, if you go back in time … Read more