Let me first define the distinction between samples and populations, as well as the notion of an infinite population. You will remember from my introductory post that one way to view the probability distribution of a random variable is as the theoretical limit of its relative frequency distribution (as the number of repetitions approaches infinity). But when working with infinite populations, things are slightly different. In short, a probability distribution is simply taking the whole probability mass of a random variable and distributing it across its possible outcomes. And more importantly, the difference between finite and infinite populations. TO BE THEIR CORRESPONDING PROBABILITY OF OCCURENCE.AM I CORRECT IN MY APPROACH ? Technically, even 1 element could be considered a sample. To get a better intuition, let’s use the discrete formula to calculate the variance of a probability distribution. Finite collections include populations with finite size and samples of populations. The set includes 6 numbers, so the denominator should be 6 rather than 5 (including in the k/5 fraction). Now let’s use this to calculate the mean of an actual distribution. Your email address will not be published. They are born out of a hypothetical infinite repetition of a random process. And that the mean and variance of a probability distribution are essentially the mean and variance of that infinite population. Notice that by doing so you obtain a new random variable Y which has different elements in its sample space. Or are the values always 1, 2, 3, 4, 5, 6, 7? From the get-go, let me say that the intuition here is very similar to the one for means. Your email address will not be published. Population mean: Population variance: For example, if you’re measuring the heights of randomly selected students from some university, the sample is the subset of students you’ve chosen. An infinite population is simply one with an infinite number of members. THANK YOU IN ADVANCE FOR YOUR CONSIDERATION ! For example, a tree can’t have a negative height, so negative real numbers are clearly not in the sample space. But how do we calculate the mean or the variance of an infinite sequence of outcomes? But where infinite populations really come into play is when we’re talking about probability distributions. Let’s compare it to the formula for the mean of a finite collection: Again, since N is a constant, using the distributive property, we can put the 1/N inside the sum operator. Let’s take a final look at these formulas. Could you give some more detail? For an arbitrary function g(x), the mean and variance of a function of a discrete random variable X are given by the following formulas: Anyway, I hope you found this post useful. And naturally it has an underlying probability distribution. The important thing is for all members of the sample to also be members of the wider population. In my previous posts I gave their respective formulas. I wrote a short code that generates 250 random rolls and calculates the running relative frequency of each outcome and the variance of the sample after each roll. Now, imagine taking the sample space of a random variable X and passing it to some function. The variance of a probability distribution is the theoretical limit of the variance of a sample of the distribution, as the sample’s size approaches infinity. Then, each term will be of the form . Is it that you have a random variable which can take on values from the set of positive integers and you generate multiple values from it? Let’s get a quick reminder about the latter. Enter a probability distribution table and this calculator will find the mean, standard deviation and variance. Filed Under: Probability Distributions Tagged With: Expected value, Law of large numbers, Mean, Probability density, Probability mass, Variance, SPYRIDON MARKOU MATLIS M.Ed. The mean of a probability distribution is nothing more than its expected value. The mean of a probability distribution is nothing but its expectation. Well, in this case they all have a probability of 1/6, so we can just use the distributive property: So, the variance of this probability distribution is approximately 2.92. Variance of a random variable can be defined as the expected value of the square of the difference between the random variable and the mean. I tried to give the intuition that, in a way, a probability distribution represents an infinite population of values drawn from it. The variance formula for a collection with N values is: And here’s the formula for the variance of a discrete probability distribution with N possible values: Do you see the analogy with the mean formula? Using the distributive property of multiplication over addition, an equivalent way of expressing the left-hand side is: That is, you take each unique value in the collection and multiply it by a factor of k / 6, where k is the number of occurrences of the value. Namely, I want to talk about the measures of central tendency (the mean) and dispersion (the variance) of a probability distribution. So, the 6 terms are: Now we need to multiply each of the terms by the probability of the corresponding value and sum the products. The Variance of a random variable X is also denoted by σ;2 but when sometimes can be written as Var (X). Since you originally operate with the actual values, couldn’t you calculate their probabilities directly? Well, intuitively speaking, the mean and variance of a probability distribution are simply the mean and variance of a sample of the probability distribution as the sample size approaches infinity. I think there’s an error in the “The mean of a probability distribution” section. In the subsequent section (“The mean and the expected value of a distribution are the same thing”), 3/5+2/5+1/5 doesn’t actually equal 1, but it would if the 5 were a 6. To get an intuition about this, let’s do another simulation of die rolls. I TAKE A SET OF VARIABLES IN AN ASCENDING NUMERICAL VALUE AND I ADD THEM UP FROM THE MINIMUM TO THE MAXIMUM VALUE SO THAT I GET THE SUM OF A SUM : Although this topic is outside the scope of the current post, the reason is that the above integral doesn’t converge to 1 for some probability density functions (it diverges to infinity). What if the possible values of the random variable are only a subset of the real numbers? Hence, we reach an important insight! You will roll a regular six-sided die with sides labeled 1, 2, 3, 4, 5, and 6. Feel free to check out my post on zero probabilities for some intuition about it. And if we keep generating values from a probability density function, their mean will be converging to the theoretical mean of the distribution. The bottom line is that, as the relative frequency distribution of a sample approaches the theoretical probability distribution it was drawn from, the variance of the sample will approach the theoretical variance of the distribution. The maximum size of a sample is clearly the size of the population. Therefore, we can say that a frequency distribution is a distribution where the total frequency is distributed over the different values of the variable in the distribution. In other words, they are the theoretical expected mean and variance of a sample of the probability distribution, as the size of the sample approaches infinity. Even the number of atoms in the observable universe is a finite number. It’s also important to note that whether a collection of values is a sample or a population depends on the context. In other words, the mean of the distribution is “the expected mean” and the variance of the distribution is “the expected variance” of a very large sample of outcomes from the distribution. Hie, you guys go to great lengths to make things as clear as possible. To conclude this post, I want to show you something very simple and intuitive that will be useful for you in many contexts. Let’s use the notation f(x) for the probability density function (here x stands for height). The association between outcomes and their monetary value would be represented by a function. This is a bonus post for my main post on the binomial distribution. If the sample grows to sizes above 1 million, the sample mean would be extremely close to 3.5. For example, if you’re only interested in investigating something about students from University X, then the students of University X comprise the entirety of your population. This section was added to the post on the 7th of November, 2020. If you repeat the drawing process M times, by the law of large numbers we know that the relative frequency of each of three values will be approaching k / 6 as M approaches infinity. All this formula says is that to calculate the mean of N values, you first take their sum and then divide by N (their number). Hi Mansoor! The moment of inertia of a cloud of n points with a covariance matrix of $${\displaystyle \Sigma }$$ is given by Infinite populations are more of a mathematical abstraction.