# Let X And Y Be Jointly Distributed Discrete Random Variables Pdf

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*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*

- AnalystPrep
- ECE600 F13 Joint Distributions mhossain - Rhea
- ECE600 F13 Joint Distributions mhossain - Rhea
- Joint distributions and independence

So far, our attention in this lesson has been directed towards the joint probability distribution of two or more discrete random variables. Now, we'll turn our attention to continuous random variables. Along the way, always in the context of continuous random variables, we'll look at formal definitions of joint probability density functions, marginal probability density functions, expectation and independence.

## AnalystPrep

These ideas are unified in the concept of a random variable which is a numerical summary of random outcomes. Random variables can be discrete or continuous. A basic function to draw random samples from a specified set of elements is the function sample , see? We can use it to simulate the random outcome of a dice roll. The cumulative probability distribution function gives the probability that the random variable is less than or equal to a particular value. For the dice roll, the probability distribution and the cumulative probability distribution are summarized in Table 2.

We can easily plot both functions using R. For the cumulative probability distribution we need the cumulative probabilities, i. These sums can be computed using cumsum. The set of elements from which sample draws outcomes does not have to consist of numbers only. The result of a single coin toss is a Bernoulli distributed random variable, i.

Note that the order of the outcomes does not matter here. We denote this as. This may be computed by providing a vector as the argument x in our call of dbinom and summing up using sum.

The probability distribution of a discrete random variable is nothing but a list of all possible outcomes that can occur and their respective probabilities. The expected value of a random variable is, loosely, the long-run average value of its outcomes when the number of repeated trials is large.

For a discrete random variable, the expected value is computed as a weighted average of its possible outcomes whereby the weights are the related probabilities.

This is formalized in Key Concept 2. This can be easily calculated using the function mean which computes the arithmetic mean of a numeric vector. To allow you to reproduce the results of computations that involve random numbers, we will used set. You should check that it actually works: set the seed in your R session to 1 and verify that you obtain the same three random numbers!

Sequences of random numbers generated by R are pseudo-random numbers, i. Since this approximation is good enough for our purposes we refer to pseudo-random numbers as random numbers throughout this book. Generally, this value is the previous number generated by the PRNG. However, the first time the PRNG is used, there is no previous value. Each seed value will correspond to a different sequence of values.

In R a seed can be set using set. Eyeballing the numbers does not reveal much. We find the sample mean to be fairly close to the expected value.

This result will be discussed in Chapter 2. Other frequently encountered measures are the variance and the standard deviation. Both are measures of the dispersion of a random variable. The variance as defined in Key Concept 2. Instead we have the function var which computes the sample variance.

The difference becomes clear when we look at our dice rolling example. The sample variance as computed by var is an estimator of the population variance. You may check this using the widget below.

Since a continuous random variable takes on a continuum of possible values, we cannot use the concept of a probability distribution as used for discrete random variables.

Instead, the probability distribution of a continuous random variable is summarized by its probability density function PDF. The cumulative probability distribution function CDF for a continuous random variable is defined just as in the discrete case. Hence, the CDF of a continuous random variables states the probability that the random variable is less than or equal to a particular value.

Due to continuity, we use integrals instead of sums. We thus have. However, this was tedious and, as we shall see, an analytic approach is not applicable for some PDFs, e. Luckily, R also enables us to easily find the results derived above. The tool we use for this is the function integrate.

First, we have to define the functions we want to calculate integrals for as R functions, i. By default, integrate prints the result along with an estimate of the approximation error to the console. However, the outcome is not a numeric value one can readily do further calculation with. Therefore we will discuss some core R functions that allow to do calculations involving densities, probabilities and quantiles of these distributions.

Every probability distribution that R handles has four basic functions whose names consist of a prefix followed by a root name. As an example, take the normal distribution. The root name of all four functions associated with the normal distribution is norm.

The four prefixes are. Thus, for the normal distribution we have the R functions dnorm , pnorm , qnorm and rnorm. The probably most important probability distribution considered here is the normal distribution.

This is not least due to the special role of the standard normal distribution and the Central Limit Theorem which is to be treated shortly. Normal distributions are symmetric and bell-shaped. The normal distribution has the PDF. In R , we can conveniently obtain densities of normal distributions using the function dnorm. Let us draw a plot of the standard normal density function using curve together with dnorm. We could use dnorm for this but it is much more convenient to rely on pnorm.

We can also use R to calculate the probability of events associated with a standard normal variate. There is no analytic solution to the integral above. Fortunately, R offers good approximations. The first approach makes use of the function integrate which allows to solve one-dimensional integration problems using a numerical method. For this, we first define the function we want to compute the integral of as an R function f. In our example, f is the standard normal density function and hence takes a single argument x.

Next, we call integrate on f and specify the arguments lower and upper , the lower and upper limits of integration. A second and much more convenient way is to use the function pnorm , the standard normal cumulative distribution function. Thanks to R , we can abandon the table of the standard normal CDF found in many other textbooks and instead solve this fast by using pnorm. R functions that handle the normal distribution can perform the standardization.

Attention : the argument sd requires the standard deviation, not the variance! An extension of the normal distribution in a univariate setting is the multivariate normal distribution. Equation 2. It is somewhat hard to gain insights from this complicated expression.

The widget below provides an interactive three-dimensional plot of 2. By moving the cursor over the plot you can see that the density is rotationally invariant, i.

The normal distribution has some remarkable characteristics. All elements adjust accordingly as you vary the parameters. The chi-squared distribution is another distribution relevant in econometrics. It is often needed when testing special types of hypotheses frequently encountered when dealing with regression models. For example, for. Further we adjust limits of both axes using xlim and ylim and choose different colors to make both functions better distinguishable. The plot is completed by adding a legend with help of legend.

As expectation and variance depend solely! At last, we add a legend that displays degrees of freedom and the associated colors. Then it holds that. The quantity. This can be computed with help of the function pf. By setting the argument lower. We can visualize this probability by drawing a line plot of the related density and adding a color shading with polygon.

Preface 1 Introduction 1. Computation of Heteroskedasticity-Robust Standard Errors 5. Part I Introduction to Econometrics with R. This book is in Open Review.

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## ECE600 F13 Joint Distributions mhossain - Rhea

Sheldon H. Stein, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor. Abstract Three basic theorems concerning expected values and variances of sums and products of random variables play an important role in mathematical statistics and its applications in education, business, the social sciences, and the natural sciences. A solid understanding of these theorems requires that students be familiar with the proofs of these theorems.

For both discrete and continuous random variables we If X and Y are discrete, this distribution can be If we are given a joint probability distribution for X under the surface that is above the region A equal to 1. x y f(x,y) x y f(x,y). Not a pdf.

## ECE600 F13 Joint Distributions mhossain - Rhea

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? As we learned in our previous lesson, there are times when it is desirable to record the outcomes of random variables simultaneously. So, if X and Y are two random variables, then the probability of their simultaneous occurrence can be represented as a Joint Probability Distribution or Bivariate Probability Distribution. Well, it has everything to do with what is the difference between discrete and continuous.

### Joint distributions and independence

Back to all ECE notes. Slectures by Maliha Hossain. We will now define similar tools for the case of two random variables X and Y. Note that we could draw the picture this way:. Note also that if X and Y are defined on two different probability spaces, those two spaces can be combined to create S,F ,P.

Back to all ECE notes. Slectures by Maliha Hossain. We will now define similar tools for the case of two random variables X and Y. Note that we could draw the picture this way:. Note also that if X and Y are defined on two different probability spaces, those two spaces can be combined to create S,F ,P. An important case of two random variables is: X and Y are jointly Gaussian if their joint pdf is given by. Find the probability that X,Y lies within a distance d from the origin.

Let X and Y be jointly continuous random variables with joint PDF fX,Y(x We know that given X=x, the random variable Y is uniformly distributed on [−x,x].

#### 1. Introduction

Sometimes certain events can be defined by the interaction of two measurements. These types of events that are explained by the interaction of the two variables constitute what we call bivariate distributions. When put simply, bivariate distribution means the probability that a certain event will occur when there are two independent random variables in a given scenario. A case where you have two bowls and each is carrying different types of candies. When you take one cady from each bowl, it gives you two independent random variables, that is, the two different candies. The fact that you are taking one candy from each bowl at the same time, you have a bivariate distribution when you are calculating for the probability of ending up with a particular kind of candies. Bivariate distribution is also referred to as Joint probability distribution and defined as the probability distribution of two random variables, X and Y defining the simultaneous behavior between the two random variables.

In this chapter we consider two or more random variables defined on the same sample space and discuss how to model the probability distribution of the random variables jointly. We will begin with the discrete case by looking at the joint probability mass function for two discrete random variables. Note that conditions 1 and 2 in Definition 5. Consider again the probability experiment of Example 3. Given the joint pmf, we can now find the marginal pmf's.

## 5 Comments

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

TedrolahandThe joint probability distribution of the x, y and z components of Random Variables. Joint probability mass functions: Let X and Y be discrete random vari- Joint PDF and Joint CDF: Suppose that X and Y are continuous random variables.

Tingvesive1978random variables are said to be jointly distributed. This section is con- If X and Y are both discrete random variables, we define the joint proba- bility mass function of X and Y Let X and Y be random variables with joint pdf. fXY (x, y) = { 1. 4.

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