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In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.

For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from 0 to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem).

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Transcription

Welcome to the presentation on moments. So just if you were wondering, I have already covered moments. You just may not have recognized it, because I covered it in mechanical advantage and torque. But I do realize that when I covered it in mechanical advantage and torque, I think I maybe over-complicated it. And if anything, I didn't cover some of the most basic moment to force problems that you see in your standards physics class, especially physics classes that aren't focused on calculus or going to make you a mechanical engineer the very next year. So we did that with-- why did I write down the word "mechanical?" Oh yeah, mechanical advantage. If you do a search for mechanical advantage, I cover some things on moments and also on torque. So what is moment of force? Well, it essentially is the same thing as torque. It's just another word for it. And it's essentially force times the distance to your axis of rotation. What do I mean by that? Let me take a simple example. Let's say that I have a pivot point here. Let's say I have some type of seesaw or whatever. There's a seesaw. And let's say that I were to apply some force here and the forces that we care about-- this was the exact same case with torque, because there's essentially the same thing. The forces we care about are the forces that are perpendicular to the distance from our axis of rotation. So, in this case, if we're here, the distance from our axis of rotation is this. That's our distance from our axis of rotation. So we care about a perpendicular force, either a force going up like that or a force going down like that. Let's say I have a force going up like that. Let's call that F, F1, d1. So essentially, the moment of force created by this force is equal to F1 times d1, or the perpendicular force times the moment arm distance. This is the moment arm distance. That's also often called the lever arm, if you're talking about a simple machine, and I think that's the term I used when I did a video on torque: moment arm. And why is this interesting? Well, first of all, this force times distance, or this moment of force, or this torque, if it has nothing balancing it or no offsetting moment or torque, it's going to cause this seesaw in this example to rotate clockwise, right? This whole thing, since it's pivoting here, is going to rotate clockwise. The only way that it's not going to rotate clockwise is if I have something keep-- so right now, this end is going to want go down like that, and the only way that I can keep it from happening is if I exert some upward force here. So let's say that I exert some upward force here that perfectly counterbalances, that keeps this whole seesaw from rotating. F2, and it is a distance d2 away from our axis of rotation, but it's going in a counterclockwise direction, so it wants to go like that. So the Law of Moments essentially tells us, and we learned this when we talked about the net torque, that this force times this distance is equal to this force times this distance. So F1 d1 is equal to F2 d2, or if you subtract this from both sides, you could get F2 d2 minus F1 d1 is equal to 0. And actually, this is how we dealt with it when we talked about torque. Because just the convention with torque is if we have a counterclockwise rotation, it's positive, and this is a counterclockwise rotation in the example that I've drawn here. And if we have a clockwise rotation, it has a negative torque, and that's just the convention we did, and that's because torque is a pseudovector, but I don't want to confuse right now. What you'll see is that these moment problems are actually quite, quite straightforward. So let's do a couple. It always becomes a lot easier when you do a problem, except when you try to erase things with green. So let's say that -- let me plug in real numbers for these values. Let me erase all of this. Let me just erase everything. There you go. All right, let me draw a lever arm again. So what we learned when we learned about torque is that an object won't rotate if the net torque, the sum of all the torques around it, are zero, and we're going to apply essentially that same principle here. So let's do it with masses, because I think that helps explain a lot of things and makes this seesaw example a little bit more tangible. Let's say I have a 5-kilogram mass here, and let's say that gravity is 10 meters per second squared. So what is the downward force here? What is the downward force? It's going to be the mass times acceleration, so it's going to be 50 Newtons. And let's say that the distance, the moment arm distance or the lever arm distance here, let's say that this distance right here is 10 meters. Let's say that I have another mass. Let's say it's a 25 kilogram-- no, that's too much. Let's say it's 10 kilograms. Let's say I have a 10-kilogram mass. And I want to place it some distance d from the fulcrum or from the axis of rotation so that it completely balances this 5-kilogram mass. So how far from the axis of rotation do I put this 10-kilogram mass? This is the distance, right? Because we actually carry the distance to the center of the mass. Well, how much force is this 10-kilogram mass exerting downwards? Well, it's 10 kilograms times 10 meters per second squared, it's 100 Newtons. This is acting what? This is acting clockwise, right? This one's acting clockwise and this one's acting counterclockwise, right? So they are offsetting each other. So we could do it a couple of ways. We could say that 50 Newtons, the moment in the counterclockwise direction, 50 Newtons times 10 meters, in order for this thing to not rotate has to be equal to the moment in the clockwise direction. And so the moment in the clockwise direction is equal to 100 Newtons times some distance, let's call that d, 100 Newtons times d, and then we could just solve for d, right? We get 50 times 10 is 500. 500 Newton-meters is equal to 100 Newtons times d. That's 100. Divide both sides by 100, you get 5 meters is equal to d. So d is equal to 5. That's interesting. And I think this kind of confirms your intuition from playing at the playground that you can put a heavier weight closer to the axis of rotation to offset a light weight that's further away. Or the other way to put it is you could put a light weight further away and you kind of get a mechanical advantage in terms of offsetting the heavier weight. So let's do a more difficult problem. I think the more problems we do here, the more sense everything will make. So let's say that we have a bunch of masses. Actually, let's not do it with masses. Let's just do it with forces because I want to complicate the issue. So this is the pivot. And let's say I have a force here that's 10 Newtons going in the clockwise direction, and let's say it is at-- let's say if this is 0, let's say that this is at minus 8, so this distance is 8, right? Let's say that I have another force going down at 5 Newtons. And let's say that its x-coordinate is minus 6. Let's say I have another force that's going up here, and let's say that it is 50 Newtons. This might get complicated. 50 Newtons, and it's at minus 2, so this distance right here is 2. Let's say that I need to figure out-- and I'm making this up on the fly. Let's say that I have another force here that is 5 Newtons. No, let's make it a weird number, 6 Newtons, and this distance right here is 3 meters. And let's say that I need to figure out what force I need to apply here upwards or downwards-- I actually don't know, because I'm doing this on the fly-- to make sure that this whole thing doesn't rotate. So to make sure this whole thing doesn't rotate, essentially what we have to say is that all of the counterclockwise moments or all of the counter clockwise torques have to offset all of the clockwise torques. And notice, they're not all on the same side. So what are all of the things that are acting in the counterclockwise direction? So counter clockwise is that way, right? So this is acting counterclockwise, this is acting counterclockwise, and that's it, right? So the other ones are clockwise. And we don't know this one. Let's assume for a second. We could assume either way. And if we get a negative, that means it's the opposite. So let's assume that this is a-- all of the clockwise ones I'll do in this dark brown. Let's assume this is clockwise, let's assume that this is clockwise, and let's assume that our mystery force is also clockwise. All of the counterclockwise moments have to offset all the clockwise moments. So what are the counterclockwise moments? Well, this one's counterclockwise, so it's 10 Newtons, 10 times its distance from its moment arm. We said it's 8, because it's at the x-coordinate minus 8 from 0, so it's 10 times 8, plus 50. This is also counterclockwise times 6, 50 times 6, and those are all of our counterclockwise moments and that has to equal the clockwise moments. So clockwise moments, let's see. We have 5 Newtons that's going clockwise times 6. 5 Newtons. Actually, was this 6? No, if this is 6, I must have written some other number here that I can't read now. How far did I say this was? Let's say that this is 2. So that 50, let's say this is 2, it's negative 2, because that's what it looks like. I apologize for confusing you. So what were all the counterclockwise moments? This 10 Newtons times its distance 8, the 50 Newtons times this distance, 2. Don't get confused by the negative. I just kind of said we're in the x-coordinate axis or at minus 8 if this is 0, but it's 8 units away, right? And this 50, its moment arm distance is 2 units. So that has to equal all of the clockwise moments. So the clockwise moments is 5 Newtons times 6. Its distance is 6 and it's 5 Newtons going in the clockwise direction. And then we have plus 6 Newtons times 3, plus 6 times 3. And then we're just assuming, we don't know for sure. Let's say we're applying the force. I should have told you ahead of time so you could do this problem. Let's say that we're applying the force at 10 meters away from our fulcrum arm. So force times 10. So now let's just solve for the force. We get 80 plus 100 is equal to 30 plus 18 plus 10F. We get 180 is equal to 48 plus 10F. What's 180 minus 48? It's 132 is equal to 10F, or we get F is equal to 13.2 Newtons. So we guessed correctly that this is going to be a-- sorry, this is going to be a-- I keep mixing up all of the clockwise and counterclockwise. This is going to be a clockwise force. These were all of the-- sorry, this is going to be a counterclockwise force, right? A clock, this is counterclockwise. Let me label that because I think I said it wrong several times in the video. So these go clockwise. And it's this one and this one. And what were the counterclockwise? These go counterclockwise. So we have to apply a 13.10 Newton force 10 meters away, which will generate 132 Newton-meters moment in the counterclockwise direction, which will perfectly offset all of the other moments, and our lever will not move. Anyway, I might have confused you with all the counterclockwise/clockwise. But just keep in mind that all the moments in one rotational direction have to offset all the moments in the other rotational direction. All a moment is is the force times the distance from the fulcrum arm, so force times distance from fulcrum arm. I'll see you in the next video.

Significance of the moments

The n-th moment of a real-valued continuous function f(x) of a real variable about a value c is

$\mu _{n}=\int _{-\infty }^{\infty }(x-c)^{n}\,f(x)\,\mathrm {d} x.$ It is possible to define moments for random variables in a more general fashion than moments for real values—see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with c = 0.

For the second and higher moments, the central moment (moments about the mean, with c being the mean) are usually used rather than the moments about zero, because they provide clearer information about the distribution's shape.

Other moments may also be defined. For example, the n-th inverse moment about zero is $\operatorname {E} \left[X^{-n}\right]$ and the n-th logarithmic moment about zero is $\operatorname {E} \left[\ln ^{n}(X)\right].$ The n-th moment about zero of a probability density function f(x) is the expected value of Xn and is called a raw moment or crude moment. The moments about its mean μ are called central moments; these describe the shape of the function, independently of translation.

If f is a probability density function, then the value of the integral above is called the n-th moment of the probability distribution. More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the n-th moment of the probability distribution is given by the Riemann–Stieltjes integral

$\mu '_{n}=\operatorname {E} \left[X^{n}\right]=\int _{-\infty }^{\infty }x^{n}\,\mathrm {d} F(x)\,$ where X is a random variable that has this cumulative distribution F, and E is the expectation operator or mean.

When

$\operatorname {E} \left[\left|X^{n}\right|\right]=\int _{-\infty }^{\infty }\left|x^{n}\right|\,\mathrm {d} F(x)=\infty ,$ then the moment is said not to exist. If the n-th moment about any point exists, so does the (n − 1)-th moment (and thus, all lower-order moments) about every point.

The zeroth moment of any probability density function is 1, since the area under any probability density function must be equal to one.

Significance of moments (raw, central, normalised) and cumulants (raw, normalised), in connection with named properties of distributions
Moment
ordinal
Moment Cumulant
Raw Central Normalised Raw Standardised
1 Mean 0 0 Mean N/A
2 Variance 1 Variance 1
3 Skewness Skewness
4 (Non-excess or historical) kurtosis Excess kurtosis
5 Hyperskewness
6 Hypertailedness
7+

Mean

The first raw moment is the mean, usually denoted $\mu \equiv \operatorname {E} [X].$ Variance

The second central moment is the variance. The positive square root of the variance is the standard deviation $\sigma \equiv \left(\operatorname {E} \left[(x-\mu )^{2}\right]\right)^{\frac {1}{2}}.$ Normalised moments

The normalised n-th central moment or standardised moment is the n-th central moment divided by σn; the normalised n-th central moment of the random variable X is ${\frac {\mu _{n}}{\sigma ^{n}}}={\frac {\operatorname {E} \left[(X-\mu )^{n}\right]}{\sigma ^{n}}}.$ These normalised central moments are dimensionless quantities, which represent the distribution independently of any linear change of scale.

For an electric signal, the first moment is its DC level, and the 2nd moment is proportional to its average power.

Skewness

The third central moment is the measure of the lopsidedness of the distribution; any symmetric distribution will have a third central moment, if defined, of zero. The normalised third central moment is called the skewness, often γ. A distribution that is skewed to the left (the tail of the distribution is longer on the left) will have a negative skewness. A distribution that is skewed to the right (the tail of the distribution is longer on the right), will have a positive skewness.

For distributions that are not too different from the normal distribution, the median will be somewhere near μγσ/6; the mode about μγσ/2.

Kurtosis

The fourth central moment is a measure of the heaviness of the tail of the distribution, compared to the normal distribution of the same variance. Since it is the expectation of a fourth power, the fourth central moment, where defined, is always nonnegative; and except for a point distribution, it is always strictly positive. The fourth central moment of a normal distribution is 3σ4.

The kurtosis κ is defined to be the normalised fourth central moment minus 3 (Equivalently, as in the next section, it is the fourth cumulant divided by the square of the variance). Some authorities do not subtract three, but it is usually more convenient to have the normal distribution at the origin of coordinates. If a distribution has heavy tails, the kurtosis will be high (sometimes called leptokurtic); conversely, light-tailed distributions (for example, bounded distributions such as the uniform) have low kurtosis (sometimes called platykurtic).

The kurtosis can be positive without limit, but κ must be greater than or equal to γ2 − 2; equality only holds for binary distributions. For unbounded skew distributions not too far from normal, κ tends to be somewhere in the area of γ2 and 2γ2.

The inequality can be proven by considering

$\operatorname {E} \left[\left(T^{2}-aT-1\right)^{2}\right]$ where T = (Xμ)/σ. This is the expectation of a square, so it is non-negative for all a; however it is also a quadratic polynomial in a. Its discriminant must be non-positive, which gives the required relationship.

Mixed moments

Mixed moments are moments involving multiple variables.

Some examples are covariance, coskewness and cokurtosis. While there is a unique covariance, there are multiple co-skewnesses and co-kurtoses.

Higher moments

High-order moments are moments beyond 4th-order moments. As with variance, skewness, and kurtosis, these are higher-order statistics, involving non-linear combinations of the data, and can be used for description or estimation of further shape parameters. The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality. This is due to the excess degrees of freedom consumed by the higher orders. Further, they can be subtle to interpret, often being most easily understood in terms of lower order moments – compare the higher derivatives of jerk and jounce in physics. For example, just as the 4th-order moment (kurtosis) can be interpreted as "relative importance of tails versus shoulders in causing dispersion" (for a given dispersion, high kurtosis corresponds to heavy tails, while low kurtosis corresponds to broad shoulders), the 5th-order moment can be interpreted as measuring "relative importance of tails versus center (mode, shoulders) in causing skew" (for a given skew, high 5th moment corresponds to heavy tail and little movement of mode, while low 5th moment corresponds to more change in shoulders).

Transformation of center

Since:

$(x-b)^{n}=(x-a+a-b)^{n}=\sum _{i=0}^{n}{n \choose i}(x-a)^{i}(a-b)^{n-i}$ where ${\dbinom {n}{i}}$ is the binomial coefficient, it follows that the moments about b can be calculated from the moments about a by:

$E\left[(x-b)^{n}\right]=\sum _{i=0}^{n}{n \choose i}E\left[(x-a)^{i}\right](a-b)^{n-i}$ Cumulants

The first raw moment and the second and third unnormalized central moments are additive in the sense that if X and Y are independent random variables then

{\begin{aligned}m_{1}(X+Y)&=m_{1}(X)+m_{1}(Y)\\\operatorname {Var} (X+Y)&=\operatorname {Var} (X)+\operatorname {Var} (Y)\\\mu _{3}(X+Y)&=\mu _{3}(X)+\mu _{3}(Y)\end{aligned}} (These can also hold for variables that satisfy weaker conditions than independence. The first always holds; if the second holds, the variables are called uncorrelated).

Sample moments

For all k, the k-th raw moment of a population can be estimated using the k-th raw sample moment

${\frac {1}{n}}\sum _{i=1}^{n}X_{i}^{k}$ applied to a sample X1, …, Xn drawn from the population.

It can be shown that the expected value of the raw sample moment is equal to the k-th raw moment of the population, if that moment exists, for any sample size n. It is thus an unbiased estimator. This contrasts with the situation for central moments, whose computation uses up a degree of freedom by using the sample mean. So for example an unbiased estimate of the population variance (the second central moment) is given by

${\frac {1}{n-1}}\sum _{i=1}^{n}\left(X_{i}-{\bar {X}}\right)^{2}$ in which the previous denominator n has been replaced by the degrees of freedom n − 1, and in which ${\bar {X}}$ refers to the sample mean. This estimate of the population moment is greater than the unadjusted observed sample moment by a factor of ${\tfrac {n}{n-1}},$ and it is referred to as the "adjusted sample variance" or sometimes simply the "sample variance".

Problem of moments

The problem of moments seeks characterizations of sequences { μn : n = 1, 2, 3, ... } that are sequences of moments of some function f.

Partial moments

Partial moments are sometimes referred to as "one-sided moments." The n-th order lower and upper partial moments with respect to a reference point r may be expressed as

$\mu _{n}^{-}(r)=\int _{-\infty }^{r}(r-x)^{n}\,f(x)\,\mathrm {d} x,$ $\mu _{n}^{+}(r)=\int _{r}^{\infty }(x-r)^{n}\,f(x)\,\mathrm {d} x.$ Partial moments are normalized by being raised to the power 1/n. The upside potential ratio may be expressed as a ratio of a first-order upper partial moment to a normalized second-order lower partial moment. They have been used in the definition of some financial metrics, such as the Sortino ratio, as they focus purely on upside or downside.

Central moments in metric spaces

Let (M, d) be a metric space, and let B(M) be the Borel σ-algebra on M, the σ-algebra generated by the d-open subsets of M. (For technical reasons, it is also convenient to assume that M is a separable space with respect to the metric d.) Let 1 ≤ p ≤ ∞.

The pth central moment of a measure μ on the measurable space (M, B(M)) about a given point x0M is defined to be

$\int _{M}d\left(x,x_{0}\right)^{p}\,\mathrm {d} \mu (x).$ μ is said to have finite p-th central moment if the p-th central moment of μ about x0 is finite for some x0M.

This terminology for measures carries over to random variables in the usual way: if (Ω, Σ, P) is a probability space and X : Ω → M is a random variable, then the p-th central moment of X about x0M is defined to be

$\int _{M}d\left(x,x_{0}\right)^{p}\,\mathrm {d} \left(X_{*}\left(\mathbf {P} \right)\right)(x)\equiv \int _{\Omega }d\left(X(\omega ),x_{0}\right)^{p}\,\mathrm {d} \mathbf {P} (\omega ),$ and X has finite p-th central moment if the p-th central moment of X about x0 is finite for some x0M.