The continuous random variable with uniform point inequality measure

# The continuous random variable with uniform point inequality measure

In the previous section, we learned about discrete probability distributions. We used both probability tables and probability histograms to display these distributions.

In this section, we shift our focus from discrete to continuous random variables. We start by looking at the probability distribution of a discrete random variable and use it to introduce our first example of a probability distribution for a continuous random variable.

X is a discrete random variable, since shoe sizes can only be whole and half number values, nothing in between. For this example we will consider shoe sizes from 6. So the possible values of X are 6. Here is the probability table for X:. As is always the case for probability histograms, the area of the rectangle centered above each value is equal to the corresponding probability.

Lecture 12: Discrete vs. Continuous, the Uniform - Statistics 110

And finally, as is the case for all probability histograms, because the sum of the probabilities of all possible outcomes must add up to 1, the sums of the areas of all of the rectangles shown must also add up to 1. Now we can find the probability of shoe size taking a value in any interval just by finding the area of the rectangles over that interval. For instance, the area of the rectangles up to and including 9 shows the probability of having a shoe size less than or equal to 9.

We can find this probability area from the table by adding together the probabilities for shoe sizes 6. Here is that calculation:. Recall that for a discrete random variable like shoe size, the probability is affected by whether or not we include the end point of the interval. For example, the area — and corresponding probability — is reduced if we consider only shoe sizes strictly less than This time when we add the probabilities from the table, we exclude the probability for shoe size 9 and just add together the probabilities for shoe sizes 6. We write this probability as. Now we will make the transition from discrete to continuous random variables.

Unlike shoe size, this variable is not limited to distinct, separate values, because foot lengths can take any value over a continuous range of possibilities. In other words, foot length, unlike shoe size, can be measured as precisely as we want to measure it. For example, we can measure foot length to the nearest inch, the nearest half inch, the nearest quarter of an inch, the nearest tenth of an inch, etc. Therefore, foot length is a continuous random variable.

## What are Continuous Variables?

What happens to the probability histogram when we measure foot length with more precision? When we increase the precision of the measurement, we will have a larger number of bins in our histogram. This makes sense because each bin contains measurements that fall within a smaller interval of values. For example, if we measure foot lengths in inches, one bin will contain measurements from 6-inches up to 7-inches.Uniform probability measures are the continuous analog of equally likely outcomes.

The standard uniform model is the Uniform 0, 1 distribution corresponding to the spinner in Figure 2. Recall that the values in the picture are rounded to two decimal places, but the spinner represents an idealized model where the spinner is infinitely precise so that any real number between 0 and 1 is a possible value. Recall that the default function used to define a Symbulate RV is the identity.

Notice the number of decimal places. Remember that for a continuous variable, any value in some uncountable interval is possible.

For the Uniform 0, 1 distribution, any value in the continuous interval between 0 and 1 is a distinct possible value: 0. A histogram is the Symbulate default plot for summarizing values on a continuous scale. It is recommended that the bins all have the same width so that the ratio of the heights of two different bars is equal to the ratio of their areas. Symbulate will always produces a histogram with equal bin widths.

We see that the bars all have roughly the same height, represented by the horizontal line, and hence the same area, though there are some natural fluctuations due to the randomness in the simulation. Recall that in a uniform probability model, the probability of an event is proportional to the size length, area, volume of the set comprising the event; two events with the same size will have the same probability.

As discussed in Section 2. We can approximate the long run average value of continuous random variables in the usual way: simulate many values and average. We see that the long run average is approximately 0.

Remember that we can define a random variable by specifying its distribution. The plots show that the values are roughly uniformly distributed between and Recall from Section 2. Remember that in a histogram, area represents relative frequency. The Uniformdistribution covers a wider range of possible values than the Uniform 0, 1 distribution. Notice how the values on the vertical density axis change to compensate for the longer range on the horizontal variable axis.

Symbulate chooses the number of bins automatically, but you can set the number of bins using the bins option, e. Preface Why study probability and simulation? An Introduction to Probability and Simulation.In probability theory and statisticsthe continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions.

The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The interval can be either be closed eg. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support. The probability density function of the continuous uniform distribution is:.

The latter is appropriate in the context of estimation by the method of maximum likelihood. Also, it is consistent with the sign function which has no such ambiguity. As the distance between a and b increases, the density at any particular value within the distribution boundaries decreases.

In graphical representation of uniform distribution function [f x vs x], the area under the curve within the specified bounds displays the probability shaded area is depicted as a rectangle. The cumulative distribution function is:. The moment-generating function is: . One interesting property of the standard uniform distribution is that if u 1 has a standard uniform distribution, then so does 1- u 1. This property can be used for generating antithetic variatesamong other things.

In other words, this property is known as the inversion method where the continuous standard uniform distribution can be used to generate random numbers for any other continuous distribution.

As long as the same conventions are followed at the transition points, the probability density function may also be expressed in terms of the Heaviside step function :.

There is no ambiguity at the transition point of the sign function. Using the half-maximum convention at the transition points, the uniform distribution may be expressed in terms of the sign function as:. The mean first moment of the distribution is:. The variance second central moment is:. Let X k be the k th order statistic from this sample.

The expected value is. The probability that a uniformly distributed random variable falls within any interval of fixed length is independent of the location of the interval itself but it is dependent on the interval sizeso long as the interval is contained in the distribution's support. This distribution can be generalized to more complicated sets than intervals. This follows for the same reasons as estimation for the discrete distributionand can be seen as a very simple case of maximum spacing estimation.

This problem is commonly known as the German tank problemdue to application of maximum estimation to estimates of German tank production during World War II. The maximum likelihood estimator is given by:. The method of moments estimator is given by:. Although both the sample mean and the sample median are unbiased estimators of the midpoint, neither is as efficient as the sample mid-rangei.

In symbols. In statisticswhen a p-value is used as a test statistic for a simple null hypothesisand the distribution of the test statistic is continuous, then the p-value is uniformly distributed between 0 and 1 if the null hypothesis is true. The probabilities for uniform distribution function are simple to calculate due to the simplicity of the function form. Furthermore, generally, experiments of physical origin follow a uniform distribution eg. In the field of economics, usually demand and replenishment may not follow the expected normal distribution.

As a result, other distribution models are used to better predict probabilities and trends such as Bernoulli process. From the uniform distribution model, other factors related to lead-time were able to be calculated such as cycle service level and shortage per cycle.

It was also noted that the uniform distribution was also used due to the simplicity of the calculations. The uniform distribution is useful for sampling from arbitrary distributions. A general method is the inverse transform sampling method, which uses the cumulative distribution function CDF of the target random variable.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

It only takes a minute to sign up. I know this is duplicated but I think the question is a bit different and needs different answer. How can CDF be continuous and have derivative at each point that is not equal to zero but the probability at each point is zero? Where this is just infinitesimal. Just an example to describe the situation. Zero probability does not mean an event cannot occur!

It means the probability measure gives the event a set of outcomes a measure zero. As Aksakai's answer points out, the union of an infinite number of zero width points can form a positive width line segment and similarly, the union of an infinite number of zero probability events can form a positive probability event. Our intuition from discrete probability is that if an outcome has zero probability, then the outcome is impossible.

If the probability of drawing the ace of spades from a deck is equal to zero, it means that ace of spades is not in the deck! With continuous random variables or more generally, an infinite number of possible outcomes that intuition is flawed. It's really not a statistics question. It's a real analysis question. For instance, it's almost the same as asking "what's the width of a point in line?

This is an interesting situation though. In mathematics the line is defined as a set of points. There are certain geometric constraints on the points, so that they form a line and not a circle, for instance. However, that's not what's important.Variable is a term used to describe something that can be measured and can also vary.

The opposite of a variable is a constant. In scientific experiments, variables are used as a way to group the data together. Variables can be grouped as either discrete or continuous variables. Generally, variables are characteristics of a group of objects or events that can be measured over a number of different numerical values. Discrete variables can have only a certain number of different values between two given points. For example, in a family, there can be one, two, or three children, but there cannot be a continuous scale of 1.

Continuous variables can have an infinite number of different values between two given points. As shown above, there cannot be a continuous scale of children within a family. If height were being measured though, the variables would be continuous as there are an unlimited number of possibilities even if only looking at between 1 and 1. It is important to remember that both kinds of variables are so grouped based on the scale used to measure them and what is being measured.

In most scientific experiments, a discrete scale is used to measure both kinds of variables. Because there are an infinite amount of possibilities, this means the measurements of continuous variables are often rounded off to make the data easier to work with.

Both discrete and continuous variables can take on one of two roles in a scientific experiment. During an experiment, the scientist often wants to observe the results of changing one variable. Only one variable is often changed, as it would be difficult to determine what had caused the relevant response if multiple variables were influenced. The variable that is manipulated by the scientist is the independent variablewhile the dependent variable is the one that responds to the change.

In other words, the response of one variable is dependent on the changes to the other variable. For example, during an experiment, the amount of light shining on a plant is changed.

The amount of light would be the independent variable. To make measurements that can be repeated, the independent variable is likely to be a discrete variable, such as one hour, two hours, or three hours of light. The response of the plant, how much it grows or the direction it grows, will be the dependent variable. As the amount the plant grows can be an infinite number of results, it is a dependent continuous variable. Please enter the following code:.

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### Continuous uniform distribution

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