The basic version of R can only provide specific commands for certain functions which include the calculation of the probability mass function, distribution function, and quantiles for binomial, geometric, negative binomial, and Poisson distributions, along with commands for simulation observations.

Probability integral transform is also known as “Universality of Uniform” as it uses a standard uniform distribution of converting the random variables from any given continuous distribution. There are times that the result would be modified or extended so that the transformation would result in the standard distribution.

The formula of a conditional expectation is E [E [X │ W]]= E [X] while the formula of conditional variance is Var [X]=E [Var [X │ W]] + Var [E [X │ W]]. When adding the terms of the conditional expectation formula, it will be easy to detect that the right-hand side of the conditional variance formula is equal to the left-hand side.

The cumulative distribution function of a random variable is as used as one of the methods in describing the distribution of the random variables. the advantage of a cumulative distribution function is that it can be defined for any kind of random variable such as discrete, continuous, and mixed, unlike the probability mass function which can only be used to describe the distribution of a discrete random variable.

Simulation is the gathering of data using a model random events that would stimulate outcomes that would match a realistic outcome with the use of actual objects such as coins and cards. By using simulations in probability, the researcher would gain insight into the real world.

The distribution function called cumulative distribution function is given by the formula of Fx(x)=Pr⁡[X≤x],x∈R which satisfies the criteria of 0≤F_X (x)≤1 for all x in R lim┬(n→→∞)⁡〖F_X 〗 (x)=1 and lim┬(n→→∞)⁡〖F_X 〗 (x)=0 which. It satisfies for all in R and which implies that most of the random variables in the distribution is non-negative or would not be less than 0.

Conditioning is one of the main tools used in risk modeling. It is often the key to a neat approach to the derivation of properties and features that were considered in risk modeling. The outcome would lead to a non-random result if there is a complete specification of the condition but if the condition is random, the outcome would also be random.

Both statements are wrong because, in the first statement, the notation Fx should be described as a non-decreasing and right continuous function. The notation that is commonly used for the expected value of a random variable x is the notation Ex. The second statement is also wrong as it should be a probability density function instead of a cumulative distribution function as it is the one known as a continuous function.

The Standard deviation of a random variable is measured to know how close the random variable is from the mean. It is called the standard deviation as it represents the “average” (standard) distance (deviation) from the mean.

Lebesgue-Stieltjes Integral generalizes the idea of Henri Leon Lebesgue and Thomas Joannes Stieltjes that preserves many advantages in creating a more general measure-theoretic framework. It is associated with any function of bounded variation in the real line.

Statistical software package R is a system used in carrying out the simulations, statistical analyses, and numerical approximations of random variables. It has a variety of packages that are available wherein users can find a diversity of codes, functions, and features that are designed to be equipped with a large amount of programming and analytical tasks. It is used with the assumption that the user is familiar with how R works and with its basic commands.

Probability generating functions are tools useful for dealing with sums and limits of discrete random variables and for some stochastic processes, it also has a special role in analyzing whether a process will reach a particular state. It is often employed for their succinct description of the sequence of probabilities of a random variable and the availability of a well-developed theory of power series with non-negative coefficients.

Kurtosis is the extent to which the peak of a probability distribution deviates from the shape of a normal distribution. The coefficient of kurtosis is the average of the fourth power, the standardized deviation from the mean. In a normal population, the coefficient of kurtosis is expected to equal to 3 wherein a value greater than 3 indicates a leptokurtic distribution while a value less than 3 indicates a platykurtic distribution.

The rth central moment of a random variable X is denoted as E[(X-E[X]^r]. The third part in the central moment is called the coefficient of skewness wherein skewness is the term that describes a lack of symmetry in a frequency contribution.

The empirical distribution function is a part of the cumulative distribution function which acts as an estimate that generates points in the sample. According to the Givenko-Cantelli theorem, it converges with probability to that underlying distribution. A series of results are used to quantify the rate of the convergence of the empirical distribution function to the underlying cumulative distribution function.

Correlation is a measurement of the strength of the linear relationship between the random variables. In correlation, there would be a positive and negative correlation. A positive correlation is a relationship wherein two variables are leading in the same direction while the negative correlation is a relationship where one of the variables decreases and the other one increases or vice versa.

Probability density function, Cumulative distribution function, and Probability Mass function are functions that can all be used in getting the probability of the likelihood of occurrence of a random variable. The probability variance function is non-existing so it cannot be used in probability theory.

The law of total variance is often useful to look at a population divided into several groups so when all of the terms in an equation are all positive, then we can conclude that by conditioning theY, the variance of x would reduce on average, furthermore, we can say that the variance of a random variable is a measure of our uncertainty about the random variable.

A simulation is only useful if it is mirrored to real-life outcomes. Several steps are required in producing a useful simulation. The steps required include: describing the possible outcomes, linkage of the outcomes of random numbers, choosing a source of random numbers, choosing a random number, noting the simulated outcomes based on the random number chosen, repeat the steps 4 and 5 multiple times until the outcome has a stable pattern and lastly, analyze the simulated outcomes and report the results.

The properties of a normal distribution are:
The mean, median, and mode have all the same value
The curve in the center of the graph is symmetric
The area from the center to the right and from the center to the left is equal
The total area under the curve is 1

Statistical software package R is a system where it has a diversity of codes and functions that can be used by the users. In risk modeling, simulations, statistical analysis, and numerical approximations are all being carried out using the statistical software package R.

All the statements about the characteristics of statistics are correct. The data stated in statistics are mostly numerically expressed as it mostly shows percentage or whole numbers. The collection of data is organized in systematic order as there is a process such as finding respondents or problems then afterward gathering the data and aggregation of facts to come up with an analysis. The collection of data in statistics is being done for a determined purpose which is to come up with an analysis and result of a tally for example.

The given formula for the cumulative distribution function is Fx(x)=Pr⁡[X≤x],x∈R. The function Fx is described as non-decreasing and right continuous which satisfies 0≤F_X (x)≤1 all x in R lim┬(n→→∞)⁡〖F_X 〗 (x)=1 and lim┬(n→→∞)⁡〖F_X 〗 (x)=0. The rest of the choices are incorrect as it describes the formula of another function.

Probability is focused on analyzing the likelihood of occurrence of an event. There are two possibilities which are called a certain event and an impossible event. A certain event has a probability of 1 which implies that the likelihood that event would occur is sure. An impossible event has a probability 0 which implies that the occurrence of the event would not happen. The probabilities of all the events would just be either 0 or 1 which implies that the event would or would not occur.

According to the book of risk modeling in general principles, they had the assumption that the reader already knew about point estimations, properties of estimators, confidence intervals, and hypothesis tests. It is also assumed that the familiarity with plots such as histograms and quantile (or Q-Q) plots are known in addition to the familiarity with the empirical contribution function.

Statistics and Simulation are taken as a prerequisite in risk modeling as the knowledge about the methods and ideas will be needed. Probability is the notation for basic quantities concerning the random variable X. Statistical software package R is a system that is being used where simulations, statistical analyses, and numerical approximations are being carried out.

In a discrete random variable n taking values in N, the expectation is E[N]=∑_(k=0)^∞〖kPr(N=k). In the given expectation, various possibilities may occur: it may be infinite, it may take the value +∞ or -∞, or it may not be defined. The expectation of a non-negative random variable would just be either a finite non-negative value or +∞.

Both probability generating functions and moment generating functions are an example of transforms. Transforms are used for the calculation that involves the sum of independent random variables. Probability generating function is described as a series of presentation of probability mass function of a discrete random variable while moment generating function is used as an alternative route to analytical results instead of working directly with probability density function or cumulative distribution function.

The three most important properties of cumulative distribution function: The function Fx of cumulative distribution is a non-decreasing function, like x→−∞x→−∞, the value of FX(x)FX(x) approaches 0 (or equals 0). That is, limx→−∞FX(x)=0limx→−∞FX(x)=0. This follows in part from the fact that Pr(∅)=0Pr(∅)=0. And as x→∞x→∞, the value of FX(x)FX(x) approaches 1 (or equals 1). That is, limx→∞FX(x)=1limx→∞FX(x)=1. This follows in part from the fact that Pr(E)=1Pr(E)=1. All of the properties mentioned above are applied equally to a discrete and continuous random variable.

The cumulative distribution function is the probability that the value of a random variable would be less than or equal to x. One of its properties is its function Fx is non-decreasing which implies that the value of the variable would not be less than 0. The probability density function is the probability that the value of a random variable would equal to the value of x. since it is a continuous distribution, its probability at a single point is equal to zero which is often expressed in terms of integration between two points. Both functions apply discrete or continuous distributions to the random variable.