Continuous Probability Distribution - Essay Example

STUDENT’S NAME: INSTITUTION: INSTRUCTOR’S NAME: DATE: Summary Continuous probability distribution A continuous random variable can assume any value in the interval on the real line or in a collection of intervals. It is not possible to talk about the probability of a random variable assuming a particular value. Instead, we talk about the probability of a random variable assuming a value within a given interval. The probability of a random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2.There are three types of distribution which are the uniform probability distribution, normal probability distribution and exponential probability distribution. They form a continuous probability distribution and each one of them has a theoretical approach that explains what these probabilities are; Uniform probability distribution A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. In the theory of probability and statistics, the uniform distributions fall into a family of symmetric probability distributions in such a manner that they are used to determine the intervals of the same length. The support provided by these intervals are defined by parameters a and b whereby a are the minimum values and b are the maximum values. The abbreviation of the distribution is U(a, b). Characteristics of uniform probability distribution The first characteristic is the probability density function which is used to find out the density function of an element under calculation. A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. The uniform probability density function is: f (x) = 1/(b – a) for a < x < b = 0 elsewhere where: a = smallest value the variable can assume b = largest value the variable can assume For the uniform probability function, the expected value of x =E(x) = (a + b)/2 and the expected variance of x= Var (x) = (b - a)2/12 Let’s take an example that explains the characteristics of uniform probability function, (Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces) The density function of the uniform probability has been defined and expressed as followed; f(x) = 1/10 for 5 < x < 15 = 0 elsewhere The graph below will help in understanding how the uniform probability distribution will appear following the example above; When we analyze the graph we can conclude that there is uniformity in the variables and the reason for the graph appearance above. The graph shows that there is proportionality of the intervals used in determining the probability. The proportional values therefore will make the graph to appear uniform. Since a random variable is uniformly distributed as per the example above we therefore use the values to determine the probability. Some other characteristics of uniform probability distribution include; cumulative distribution function and generating functions. The properties of uniform probability distribution are used to further explain the functions of a continuous distribution. For some cases, the properties are used when the probability is in extreme and cannot be handled by the normal functions that define uniform probability. For example properties such as; uniformity, this is a property that explains how the distribution is assumed to be uniformly distributed, the variables will fall under any interval that is within the range of any fixed length. Generalization is another property that describes how the uniform probability distribution can be generalized in terms of the interval sets such that they can be used to define a finite measure. Normal probability distribution The normal probability distribution is the most important distribution for describing a continuous random variable. It is widely used in statistical inference. The theory behind the normal distribution explains that is a type of distribution that is symmetric in nature and that the appearance depicts its mean, it is at the peak. As the probability distribution decreases, the mean also decreases smoothly as shown in the graph below; The graph shows the impact of mean in the normal distribution such that when the mean is low, the graph peak will also be low. When the mean is high the peak of the graph will also be high. Therefore we can conclude that the mean determines the peak of the graph. In statistics, the occurrence of normal probability is frequent since almost all the variables used in statistics should have a mean. Also in economics and natural and social sciences, we find that the normal probability occurs because they involve comparison and calculation of the variables. Normal distribution is used in most areas related to nature and social sciences in the sense that distributions occur in everyday lives for example determining the height of people, we find that the distribution varies and therefore it is normal, another example is the income distribution of people in particular job groups, we find that normal distribution occurs in every aspect of human life and in order to determine their rates of distribution through statistics, we used the mean mode and median. Therefore the graph above depicts the occurrence of normal distribution. The theory used in explaining the normal distribution is also used in astronomy whereby the astronomists have been able to predict the nature appearance by the used of distribution factor. The characteristics of the normal distribution are the mean and standard deviation whereby the mean is calculated by determining the overall value of the collected variables and dividing them by the number of units under the study. The mean is used to determine the point of the peak while standard deviation will measure the spread of normal probability distribution. Normal Probability Density Function where: m = mean s = standard deviation p = 3.14159 e = 2.71828 The characteristics of normal probability distribution are defined below, each characteristic comes from the calculation and determination of a normal distribution. The distribution is symmetric and is bell-shaped this explains the curved shape of the outcome of any normal distribution results. The entire family of normal probability distributions is defined by its mean m and its standard deviation s, this explains why there should be the use of standard deviation to determine the end result of the normal distribution. The highest point on the normal curve is at the mean, which is also the median and mode. When we observe the curve of a normal distribution, we find that the mean defines the peak and therefore it forms the highest point. The mean can be any numerical value: negative, zero, or positive, therefore any graph that describes the mean will always have a peak because in calculating the mean, there should be high and low mean which could be ranging from negative to zero, then zero to positive. The standard deviation determines the width of the curve: larger values result in wider, flatter curves. Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right). A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability distribution. The letter z is used to designate the standard normal random variable. Exponential distribution The exponential probability distribution is useful in describing the time it takes to complete a task. The exponential random variables can be used to describe: time between the arrival of vehicles at the toll booth, the time required to accomplish a questionnaire and the distance between major effects in the highway. Generally, exponential probability distribution explains the time waited for the next event to occur; therefore they are used in calculating the expected occurrence of the next event. Exponential distribution is used mostly to predict the occurrence of a scenario and in many cases the function below describes the cumulative probability; where: x0 = some specific value of x The graph below shows the appearance of an exponential distribution We find that the graph depicts the occurrence of events such the appearance is a curve shape, therefore we can conclude that the occurrence of an event is dependent upon the occurrence of the first event and the same case applies to the future events. The properties that help in determining exponential distribution are; the expected value, variance, moment generating function, characteristic function and distribution function. When the time used to wait for the next event to occur is not known, we therefore consider it as a random variable and therefore refer to is as exponential distribution. Another name for exponential distribution is the poison distribution whereby occurrence of two events more than once and there is exponential distribution on the time occurrence such that there is a lapse of time, the number of occurrences have the Poisson distribution. Relationship between poison and exponential distribution The Poisson distribution provides an appropriate description of the number of occurrences per interval. This is explained by the fact that for the case of Poisson, there should be more than one occurrence of the same events and therefore it describes the numbers of occurrences per interval. For this case, it is easier to calculate the occurrence of the next event since they are measured by the interval. on the other hand, The exponential distribution provides an appropriate description of the length of the interval between occurrences. In this case the description is not exact as it is explained that it provides only appropriate results on the occurrence of the event. To further understand the theory behind the exponential distribution, let’s look at the example below; “The time between arrivals of cars at Al’s full-service gas pump follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less.” Having a look at the above example we find that the service gas pump has an already approximated time for the arrival of the next car after the departure of the previous, in this case there is a gap on what will be the time for the arrival of the next car and therefore Al would want to determine whether the next car arrival will fall between two or less minutes. The formula above on an exponential distribution will be used to calculate the next arrival of the car. Work cited Read More