Inmost cases, a probability density function is an equation that isused to describe the continuous probability distribution. The graphis continuous over the provided range of values because of the randomvariable, which, in this case, is also continuous over that range.For the density function, the area of the curve under it and thex-axis total to 1 if the computation is done regarding the domain ofthat variable. For the two points, a and b, the chance that a randomvariable can take any value between them is the same as the areacovered by the density function between the two points.
Themean and the standard deviation are basic parameters when it comes tonormal distribution. The factors are critical when in the definitionof the probability density function as explained by the text.However, when writing the probability density functions, there areimportant concepts that one must know concerning the mean and thestandard deviation. There is a great difference between the normaldistributions as they take many forms. It is critical to note thatmeans and their standard deviations can also differ depending on theway that normal distribution takes. The difference arises accordingto the side where distribution is the most. Three normaldistributions are therefore evident here. They include the left-most,the middle, and right most (Bryc, 2012). The three will not have thesame mean and standard deviation as the points to which they aredistributed is not similar. An important observation to make as wellis that the all the normal distributions are symmetric. Because ofthis reason, more values are usually found at the center of thedistribution, and a few of them in the tails. Therefore, in theprobability density function, the mean and standard deviation mightchange depending on the side where distribution is more.
Theprobability density function can be used to describe the continuousprobability distribution. The conditions for continuous probabilitydistributions hold for it to be differentiated from otherdistributions. The chance of a random variable assuming theparticular value is zero. Expressing this distribution in the tabularform is therefore not possible. This explains why the equations offormulas are the most preferred when describing the continuousprobability distribution. For the probability density function, therandom variable Y must be a function of x. For all the values of x, ycan only be greater than or equal to zero. Finally, the area underthe curve should be equal to zero.
Probabilitydistribution or assessment has a role to play before making anydecision. The evaluation is the quantification of uncertainty. Thisis what allows communication to be made concerning any two eventsthat are not known. Probability is used to communicate the risk aswell as managing it effectively. Depending on the degree of knowledgean individual has towards a given event, decisions can be maderegarding any scenario. The nature of the risk must be known. It canbe pure uncertainty, risk or buying of the information. Probabilitywill enable the decision maker to know the chances of an eventoccurring or not. After this, decisions can then be made. It isevident that the probability assessment is what quantifies theinformation gap concerning what we know and the unknown. It is,therefore, fair to say that probability distribution or assessment iswhat quantifies the lack of information on what is known and need tobe figured out before making a decision.
Bryc,W. (2012). TheNormal Distribution: Characterizations with Applications.New York: Springer Science & Business Media.