In probability theorya probability mass function abbreviated pmf gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function in that the values of the latter, defined only for continuous random variablesare not probabilities; rather, its integral over a set of possible values of the random variable is a probability.

The discontinuity of probability mass functions reflects the fact that the cumulative distribution function of a discrete random variable is also discontinuous.

Where it is differentiable i. A simple example of a probability mass function is the following. Suppose that X is the outcome of a single coin toss, assigning 0 to tails and 1 to heads. Probability mass functions may also be defined for any discrete random variable, including constantbinomial including Bernoullinegative binomialPoissongeometric and hypergeometric random variables.

**probability density functions and cumulative distribution functions s1**

Sign In Don't have an account? Examples Edit A simple example of a probability mass function is the following. Categories :. Cancel Save. This page uses Creative Commons Licensed content from Wikipedia view authors.The probability mass function pmf characterizes the distribution of a discrete random variable.

### Constructing a probability distribution for random variable

It associates to any given number the probability that the random variable will be equal to that number. In formal terms, the probability mass function of a discrete random variable is a function such that where is the probability that the realization of the random variable will be equal to.

Suppose a random variable can take only three values 1, 2 and 3each with equal probability. Its probability mass function is.

## Probability with Venn diagrams

So, for example, that is, the probability that will be equal to is. Or, that is, the probability that will be equal to is equal to. Note that the probability mass function is defined on all ofthat is, it can take as argument any real number. However, its value is equal to zero for all those arguments that do not belong to the support of i. On the contrary, the value of the pmf is positive for the arguments that belong to the support of. In the example above, the support of is As a consequence, the pmf is positive on the support and equal to zero everywhere else.

Often, probability mass functions are plotted as column charts. For example, the following plot shows the pmf of the Poisson distributionwhich is We set the parameter and plot the values of the pmf only for arguments smaller than note that the support of the distribution isi. You can find an in-depth discussion of probability mass functions in the lecture entitled Random variables. Joint probability mass function : the pmf of a random vector.

Marginal probability mass function : the pmf obtained by considering only a subset of the set of random variables forming a given random vector. Conditional probability mass function : the pmf obtained by conditioning on the realization of another random variable. Previous entry: Probability density function. Taboga, Marco Kindle Direct Publishing.

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Donate Login Sign up Search for courses, skills, and videos. Math Precalculus Probability and combinatorics Venn diagrams and the addition rule. Probability with Venn diagrams.

Addition rule for probability. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript Let's do a little bit of probability with playing cards. And for the sake of this video, we're going to assume that our deck has no jokers in it. You could do the same problems with the joker, you'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits, and the suits are the spades, the diamonds, the clubs, and the hearts.

You have four suits and then in each of those suits you have 13 different types of cards-- and sometimes it's called the rank. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, ten, and then you have the Jack, the King, and the Queen. And that is 13 cards. So for each suit you can have any of these-- you can have any of the suits. So you could have a Jack of diamonds, a Jack of clubs, a Jack of spades, or a Jack of hearts.

So if you just multiply these two things-- you could take a deck of playing cards, take out the jokers and count them-- but if you just multiply this you have four suits, each of those suits have 13 types. So you're going to have 4 times 13 cards, or you're going to have 52 cards in a standard playing deck.

### Probability mass function

Another way you could have said, look, there's 13 of these ranks, or types, and each of those come in four different suits-- 13 times 4. Once again, you would have gotten 52 cards. Now, with that of the way, let's think about the probabilities of different events.

So let's say I shuffle that deck. I shuffle it really, really well and then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a Jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards.

So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are Jacks?Review the recitation problems in the PDF file below and try to solve them on your own. One of the problems has an accompanying video where a teaching assistant solves the same problem. Review the tutorial problems in the PDF file below and try to solve them on your own. Work the problems on your own and check your answers when you're done.

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Donate Login Sign up Search for courses, skills, and videos. Constructing a probability distribution for random variable. Valid discrete probability distribution examples. Probability with discrete random variable example. Practice: Probability with discrete random variables. Mean expected value of a discrete random variable. Practice: Mean expected value of a discrete random variable. Variance and standard deviation of a discrete random variable.

Practice: Standard deviation of a discrete random variable. Mean and standard deviation of a discrete random variable.

Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript Voiceover:Let's say we define the random variable capital X as the number of heads we get after three flips of a fair coin.

So given that definition of a random variable, what we're going to try and do in this video is think about the probability distributions. So what is the probability of the different possible outcomes or the different possible values for this random variable. We'll plot them to see how that distribution is spread out amongst those possible outcomes. So let's think about all of the different values that you could get when you flip a fair coin three times.

So you could get all heads, heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, heads. You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails. So there's eight equally, when you do the actual experiment there's eight equally likely outcomes here.

But which of them, how would these relate to the value of this random variable? So let's think about, what's the probability, there is a situation where you have zero heads. So what's the probably that our random variable X is equal to zero? Well, that's this situation right over here where you have zero heads. It's one out of the eight equally likely outcomes.

What's the probability that our random variable capital X is equal to one? Well, let's see.In probability and statisticsa probability mass function PMF is a function that gives the probability that a discrete random variable is exactly equal to some value. The probability mass function is often the primary means of defining a discrete probability distributionand such functions exist for either scalar or multivariate random variables whose domain is discrete.

A probability mass function differs from a probability density function PDF in that the latter is associated with continuous rather than discrete random variables. A PDF must be integrated over an interval to yield a probability. The value of the random variable having the largest probability mass is called the mode. Probability mass function is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities.

The probabilities associated with each possible values must be positive and sum up to 1. For all other values, the probabilities need to be 0. We make this more precise below. The discontinuity of probability mass functions is related to the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Discretization is the process of converting a continuous random variable into a discrete one. There are three major distributions associated, the Bernoulli distributionthe Binomial distribution and the geometric distribution.

Other distributions that can be modeled using a probability mass function are the Categorical distribution also known as the generalized Bernoulli distribution and the multinomial distribution. Two or more discrete random variables have a joint probability mass function, which gives the probability of each possible combination of realizations for the random variables. From Wikipedia, the free encyclopedia.

Discrete-variable probability distribution. Main articles: Bernoulli distributionBinomial distributionand Geometric distribution. Main article: Joint probability distribution.

Princeton University Press. Dekking, Michel, London: Springer. New York: Wiley. CS1 maint: multiple names: authors list link. Theory of probability distributions. Categories : Types of probability distributions. Hidden categories: CS1 maint: others CS1 maint: multiple names: authors list Articles with short description Short description is different from Wikidata.

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Probabilities from density curves. Practice: Probability in density curves. Practice: Probability in normal density curves. Next lesson.

Current timeTotal duration Google Classroom Facebook Twitter. Video transcript In the last video, I introduced you to the notion of-- well, really we started with the random variable. And then we moved on to the two types of random variables. You had discrete, that took on a finite number of values. And the these, I was going to say that they tend to be integers, but they don't always have to be integers. You have discrete, so finite meaning you can't have an infinite number of values for a discrete random variable.

And then we have the continuous, which can take on an infinite number. And the example I gave for continuous is, let's say random variable x. And people do tend to use-- let me change it a little bit, just so you can see it can be something other than an x. Let's have the random variable capital Y. They do tend to be capital letters. Is equal to the exact amount of rain tomorrow. And I say rain because I'm in northern California. It's actually raining quite hard right now.

We're short right now, so that's a positive. We've been having a drought, so that's a good thing. But the exact amount of rain tomorrow. And let's say I don't know what the actual probability distribution function for this is, but I'll draw one and then we'll interpret it. Just so you can kind of think about how you can think about continuous random variables. So let me draw a probability distribution, or they call it its probability density function. And we draw like this. And let's say that there is-- it looks something like this.

Like that. All right, and then I don't know what this height is.

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