# Joint Pdf Of Discrete Random Variables And Their Probability

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*Even math majors often need a refresher before going into a finance program.*

- Joint probability distribution
- 5.1: Joint Distributions of Discrete Random Variables
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*Did you know that the properties for joint continuous random variables are very similar to discrete random variables, with the only difference is between using sigma and integrals?*

## Joint probability distribution

Even math majors often need a refresher before going into a finance program. This book combines probability, statistics, linear algebra, and multivariable calculus with a view toward finance. You can see how linear algebra will start emerging The marginal probability mass functions are what we get by looking at only one random variable and letting the other roam free.

You can think of these as collapsing back to single-variable probability. Think about how this gives the marginal probability mass functions above. Toss a quarter and a dime into the air. Definitely not — simply by reasoning about the problem, we know that how much money shows tails is related to how many coins show heads.

For a contrasting discrete probability problem, we can return to dice! Roll two dice. Using our previous rules for conditional probability, we know that. Following through the definitions, we can find a conditional probability mass function as well:.

Again, you can often use reasoning to guide you to whether two random variables are independent or not. Definition 2. Remember that in single-variable probability we were able to find the pdf from the cdf by differentiating. Now we need partial derivatives, because we have two variables to consider. Using the multivariate fundamental theorem of calculus, we can see that.

Example 2. Notice the bounds of integration here: why are they what they are? Thus the pdf is. These are both functions of a single variable, and so only that variable can appear in the expression for the marginal pdf.

If we do this, we can define conditional expected value:. Covariance and correlation were mentioned briefly earlier. Remember that we looked at the variance of a sum of independent random variables in Chapter 5.

Now, we care about non-independent random variables jointly distributed random variables! There are of course more convenient formulas. Another definition of covariance is. If you need to compute the covariance of two random variables, this is often the easiest way to do it. This is exciting! Trust me! For a random vector. Last, we can prove that covariance matrices are always positive semidefinite by using the following trick.

Covariance has the drawback of being intimately related to the units and magnitudes of the random variables under consideration. Correlation is the unitless sister to covariance, and more easily allows comparisons between different types of information.

Mentioning the Cauchy-Schwarz inequality should give you flashbacks to the section on vector inequalities Section 8. In fact, covariance is conceptually a lot like an inner product, and in particular like the dot product for real vectors.

Remember that the dot product for real vectors satisfied a few properties:. In the linear algebra sections of the book, we spent significant effort on changing bases. We can make all sorts of changes of variables! First, the technical details; then some examples. Along with the inverse of the transformation, we need to know how the transformation changes the variables infinitesimally. We understand this change through using the Jacobian,.

Determinants are intimately related to volume — remember the determinant of a three by three matrix is the volume of the parallelopiped spanned by the three columns of the matrix, for instance.

This was discussed in section 8. The combination of the determinant of the matrix of derivatives can be thought of as measuring the change of volume that is forced by the transformation. The bivariate normal distribution is a great place to start exploring multivariate normal distributions, as we can actually draw pictures. Definition 5. While a bit abstract, this is actually a very useful characterisation of bivariate normal distributions.

You might wonder, though, what the probability density function is. Then the pdf for their bivariate normal distribution is. As in the single-variable case, we can transform our way from this straightforward density function to any other bivariate normal density function.

Use the multivariate change of variables formula discussed earlier. In our current situation, we have inverse functions.

Why do I call this horrible? Because it takes a long time to type, and some people find it easy to mess up the recall of the formula on exams due to its length. Use the power of linear algebra to simplify this expression! If we write. You need to familiarize yourself with the probability density function for the bivariate and multivariate normal distributions to call yourself a financial mathematician.

For instance, you might want to do a basic calculation like this:. Example 5. Using standardization and z-tables will be a fine technique for solving many multivariate normal problems, and I would be negligent not to discuss these problems here. On the other hand, you can find this material in many probability texts and I urge you to visit them for examples see for instance Rosencrantz. At first glance this all seems a bit obvious, but the power of mathematics only comes into play if you dig deeper and question the obvious.

Why do we always get ellipses? Do we always get ellipses? If we have ellipses, what does their shape tell us? Any ideas? Well, yes and no. Why an ellipse? To answer this, we need quadratic forms. This is a quadratic form!! How do we know that? Speaking of these eigenvectors, they provide the directions of the major and minor axes of the ellipses under consideration. Joining the email list for this book will allow the author to contact you to let you know about special offers and when updates for the book are available.

Please enable javascript This site requires you to allow JavaScript to run in the browser for all features to work. Thank you! Setup Log In Sign Up. Mathematical Preparation for Finance A wild ride through mathematics Kaisa Taipale Even math majors often need a refresher before going into a finance program. Book Info. Mailing List. Single Page. Follow this book to receive email updates from the author. Chapter 10 Joint distributions. But you need measure theory for this. Sorry, content not available.

## 5.1: Joint Distributions of Discrete Random Variables

Associated to each possible value x of a discrete random variable X is the probability P x that X will take the value x in one trial of the experiment. The probability distribution A list of each possible value and its probability. The probabilities in the probability distribution of a random variable X must satisfy the following two conditions:. A fair coin is tossed twice. Let X be the number of heads that are observed. The possible values that X can take are 0, 1, and 2. The probability of each of these events, hence of the corresponding value of X , can be found simply by counting, to give.

If X and Y are discrete, this distribution can be described with a joint probability mass function. If X and Y are continuous, this distribution can be described with a joint probability density function. The possible values of Y are 15 and 16 mm (Thus, both X and Y are discrete).

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Metrics details. In this paper a comprehensive survey of the different methods of generating discrete probability distributions as analogues of continuous probability distributions is presented along with their applications in construction of new discrete distributions. The methods are classified based on different criterion of discretization.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs and how to get involved. Authors: Shahana Ibrahim , Xiao Fu.

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## 5 Comments

Jay D.If discrete random variables X and Y are defined on the same sample space S, then their joint probability mass function (joint pmf) is given by.

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Eustaquia L.These ideas are unified in the concept of a random variable which is a numerical summary of random outcomes.

Odelette A.In the case of only two random variables, this is called a bivariate distribution , but the concept generalizes to any number of random variables, giving a multivariate distribution.

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