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Probability is a broad topic which most students find it challenging tackling its questions. But as you know, passing your probability exam is of much importance as it contributes to our overall grades. To stand a better chance, you must score equally well in your assignments. Nonetheless, you don’t have to sweat with such tasks. All you have to do is to ask for our probability assignment help.

Probability theory is a branch of mathematics related to probability and analyzing random events. The central objects of probability theory are random variables, stochastic processes (especially series of random variables), and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in terms of a probability distribution. The mathematical theory of probability is used as a tool in many applied fields, including statistics, machine learning, finance, gaming, physics, data science and other sciences.

Probability theory is also employed in various attempts to rationalize seemingly chance-based events such as philosophy and religion.

For example: If you toss a coin enough times then eventually it will be heads…

Probability theory tries to predict the number of times that it will be heads. In many cases, theory and experiment agree so well that the probability falls under mild uncertainty (for example, tossing a fair coin does not exhibit any significant deviations from 50%).

Probability theory is used in mathematical finance to determine the value of derivatives, options etc.

Probability theory experiments can be carried out using a random device such as dice or an urn with marbles colored red or black, but may also take place in nature due to chaotic variables independent of human control.

Random events can be divided into 3 categories namely independent events, dependent events, or mixed events.

Independent events: An 'independent' event is one in which a probability distribution cannot be constructed based on the information on the occurrence of previous events because it requires knowledge not just about the individual events but also about their relationship with each other and even the manner in which they are generated.

Dependent events: A 'dependent' random variable is one that depends on or can only take values determined by the outcomes of an independent variable.

A roll of dice for example has no value unless there are numbers printed on them so it is impossible to tell what will happen if we throw them until they land showing their values.

Hence we say this rolle has a "mixed" probability distribution because its possible outcomes depend on (are mixed with) each other.

A "mixed" event is composed of a set of independent events.

Probability theory provides an excellent way for researchers to model uncertainty based on the understanding that many processes are random.

There is no unified theory for probability theory. The various methods listed above have been developed in order to ensure consistency with different branches of science, including chemistry and biology.

Probability is useful because it allows researchers to study a wide range of events without having to go through the lengthy process of running trial experiments over and over again. Instead, mathematicians can use the principles of probability theory to calculate the likelihood that an event will occur when given specific parameters.

The method of least squares is one way in which our online statistics tutors and online probability tutors can study the probability that an event will occur based on previous data.

What Is Probability And Its Basic Rules?

Probability is a numerical description of how likely an event is to occur. Alternatively, it could be how likely it is that a proposition is true. It ranges from 0 to 1, where 0 is an impossibility, and 1 indicates certainty.

Otherwise, different books tend to have different definitions of probability, and this, therefore, poses a challenge to students, even making it hard for them to understand it. Our Probability Assignment Help comes in handy as it simplifies everything helping you understand the basic rules of probability. These rules are:

Classical (sometimes called “A priori” or “Theoretical”) – if we have a random process in which there are “n” equally likely outcomes, and the event A consists of exactly “m” of these outcomes, we say that the probability of A is m/n, which write as “P(A)=m/n” for short. This perspective has the advantage that it is conceptually simple in many situations. However, it is limited, since many situations do not have finitely many equally likely outcomes.

Empirical - This perspective defines probability via a thought experiment. The empirical view of probability applies to most statistical inference procedures. These are called frequentist statistics. The frequentist views what gives credibility to standard estimates based on sampling.

Subjective – is an individual’s measure of belief that an event will occur. This probability fits well with Bayesian statistics, which are an alternative to the more common frequentist statistical methods.

Axiomatic - is a unifying perspective. We can prove the coherence conditions needed for subjective probability hold for the classical and empirical definitions. The axiomatic perspective codifies these coherence conditions, so, we can use it with any of the above three perspectives.

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Types of probability assignment help

• Classic probability help-This often involves coin tossing or rolling dice. It is calculated by listing all of the possible outcomes of the activity and recording the actual occurrences.
• Experimental probability help -It is based on the number of possible outcomes by the total number of trials.
• Theoretical probability help -It is an approach that bases the possible probability on the possible chances of something happening.
• Subjective probability-It is based on a person’s own personal reasoning and judgement.It is the probability that the outcome a person is expecting will actually occur. There are no formal calculations or subjective probability but instead it is based on a person’s own knowledge and feelings.

Basic properties of probability theory

In probability theory a random event is some occurrence, whose outcome is uncertain. You know the chance it will happen but you don't know for certain that it will or won't happen.

The probability function is a term that's used to describe the likelihood that an event will happen under certain circumstances.

A discrete random variable is one that can take any of a specified finite set of possible values e.g., the number of marbles in an urn (we could have 0, 1, 2, ... , 1234).

A continuous random variable on the other hand has infinitely many potential values (such as the position of a particle in space or time) and is usually characterized by its probability density function which gives the probability that this variable will take a specific value x.

Basic Probability Rule

If an event A is the union of k events, then P(A) = √P(Ai) where i is a symbol for counting integers and Ai denotes Ai as a subset. This rule makes it easy to compute probabilities of events which are composed of smaller independent events.

Law of Total Probability

The Law of Total Probability states that in order to calculate the probability of an event happening and it is also possible that it does not happen, we can add together all of the probabilities for either the event or its complement. It states: P(A∪B)=P(A)+P(B)-P(A*B).

Conditional Probability help

This rule allows you to calculate the probability that one event A will happen, given that another event B has happened. The formula is: P(A|B) = PB * P(B).

Chain Rule

This theorem states It states: if an experiment has n possible outcomes Ai then for any two of these there is some relationship between their probabilities e.g., P(Ai)=P(Bi)-P(Ai)*P(Bi|Ai). We can use this relationship to calculate the probability of any event by adding together all of the relevant conditional probabilities. This theorem allows us to apply Bayes' theorem when two events are linked.

Bayesian Probability

This rule states that if we know the frequencies with which different outcomes occur in repeated experiments and we also know the chance of each outcome given X, then P(X)=∑PiP(Xi|X) where i runs through all possible outcomes.

Complement probability rule

The complement rule replaces "not A" with "the event that is not A" so for example P(A')=1-P(A).

Conditional independence probability help

If two random variables X and Y are independent then the probability of them both happening together, P(X+Y) can be calculated by multiplying their probabilities P(X)×P(Y). If they are not independent then this formula is incorrect.

If random variables X and Y are independent then their joint probability P(X+Y) is the product of two conditional probabilities, P(X|Y)×P(Y).

Bayes Theorem help

This theorem states that if our hypothesis H1 predicts X more than are hypothesis H2 then the ratio between P(X|H1)/ P(X|H2) will increase as our evidence E increases. This allows us to use Bayesian Probability to determine what should come first when using evidence from repeated experiments, a hypothesis or a conjecture.

Probability distribution function (pdf) homework help

This function describes the chance of exactly x occurrences of an event A in an infinite number of independent trials. Probability density function , p(x) is a special case.

Central limit theorem

This theorem applies when you have a large number of independent and identically distributed observations which tend to increase with n, although not necessarily in a linear fashion. The variance tends to decrease as n increases and this causes the distribution to become increasingly 'normally' distributed.

Binomial distribution

This function describes the number of successes out of a certain amount (n) of independent trials. It can be shown that P(X)=np where p is the chance of success in one trial and n=the amount of trials.

Poisson distribution

The Poisson Distribution function can be used when there are many occurrences in a small time interval or if an event has a low probability but occurs frequently. e.g., how often will you win the lottery?

The formula is: P(X)=√λe–λ*x which means that if we know our average rate λ then we can calculate the probability that any particular x events occur within an interval by using the formula √ex.

Binomial-Geometric Distribution

This adds a geometric distribution to the binomial distribution. For example, P(X=x)=Γ(λ)×P(X=0)×e–λ*x! where λ is the average rate of occurrence and x is the number of occurrences we are interested in.

Poisson Approximation to Binomial Distribution (Skellam's Distribution)

If p≈1 this distribution approximates the binomial result for n>30. It can be shown that when N is infinite then they have equal distributions. However, if we take small values of p then the Poisson equation will return similar results however there are some subtle differences.

Binomial Distribution Approximation to Poisson Distribution in Large n

If p is small and we have large values of n then the binomial distribution can be used to approximate the Poisson equation with accuracy greater than or equal to 5% for all x≥3.

Exponential Distribution Function

This describes the time between successive occurrences of an event. If λ>0, it describes the time between events that occur at a constant average rate (e.g., how long it takes before you win a game of chance). In this case P(X=x)=λ×e–λ*x where λ is the average rate [5]. This function closely corresponds to many Probability Distributions and the related exponential function.

Gamma Distribution Function

The Gamma function is used to describe the change in a quantity over time . The Gamma distribution is often applied to errors that occur repeatedly. For example, if we have an experiment where there is an 85% chance of success then we can say with some certainty that the probability of obtaining a particular number of successes in a fixed number of trials will be normally distributed between 15% and 95%. However, if our error rate wasn't constant (85%) but varied according to some other more complex law then it may not be practical to use a normal distribution as an approximation because Γ(x) doesn't vary linearly with x.

Hazard Function

This is defined as H(t)=the probability that a failure has not occurred at time t and will occur within the next instant.

Survival Function

This function is used to describe the ratio of people who have survived past some event. S(t) describes how many successes there are after a particular interval and can be seen in the following formula: S(t)=∑xk=0∞e-λ*k where λ represents our rate of occurrence.

Weibull Distribution Function (Exponential Distribution)

The Weibull distribution function can be calculated by using this formula: W(x)=e-"m*x" if x>0 and W(x)=0 if x<0. Since the Weibull distribution function has a maximum of 1 it can be seen that "m" must be positive. Weibull probability density function is used when we need to analyze how the power of a process will change over time

Pareto Distribution Function

This is used to describe the relative number of successes in relation to some large number of events. It can be calculated with this formula: P(X>x)=Γ(a)×e-"λ*x" for λ>0 where a and λ represent parameters that we must determine. An example would be how many customers do you need to serve before at least one person becomes dissatisfied? A normal distribution could also be used as an approximation however it only describes the probability density functions in terms of x, whereas the Pareto distribution also includes a parameter "a" that describes the absolute number of successes, so it is much more flexible. The maximum value of its probability density function is 1 and that of its cumulative distribution function is 1, which makes it an extremely useful tool when comparing systems with unequal mean performance measures.

Lognormal Distribution Function

The Lognormal distribution functions are used in cases where we have information about both the mean and standard deviation values but we want to know how these relate to each other through the use of some predictive distribution function/density function.

Lambda Distribution Function

This function is used when an experimenter can have varying degrees of success or failure which means it represents the probability that something will happen at some point in time. The Lambda function has special applications in the study of economics where companies need to consider how they manage risk and uncertainty. By using real world examples it was found that practical use of this tool reduces errors by 60%.

Rayleigh Distribution Function

This function is the probability density of a continuous random variable with one parameter which represents its mean. Since Rayleigh values are often small this makes it very useful in cases where we have to predict a specific frequency or count and we just want to make sure that our results fall within some range.

Fractional Brownian Motion (FBM) Distribution Function

Since Brownian motion has many uses in physics, biology and finance research, there are several different approaches for calculating its related distribution functions.

Erlang Distribution Function

This distribution function is used when we want to know how many events will occur within a certain amount of time and it contains two parameters: alpha and beta.

Pareto Distribution Function

The Pareto distribution functions are special cases of power laws which describe how a system changes in response to an input.

Rayleigh Quotient Distribution Function

To determine the probability density function of this distribution we must calculate it by taking the ratio of two Gamma functions since it is based on how a Poisson process can be recombined in certain scenarios. This function allows us to model how an event rate will change over time and has important applications for models that describe traffic or other systems.

Thurstonian Distribution Function

This distribution function is only used in Fractal Geometry when a person is analyzing their work based on fractals that are defined by real world sets of data. Also discussed in depth in big data assignment help section.

Lanczos Distribution Function

Using this approach researchers are able to simulate many different things including:

• gene mutation rates through replication errors,
• random drift by looking at how particles move through a fluid,
• wave functions of atoms or molecules.

Plethora Distribution Function

This function is used to calculate the probability that we will see an event occur at least once in a certain time period. This concept is useful in our genetics homework help service. Plethora distributions are a type of function that can be used when you have too many random events occurring for one parameter.

Multinomial distribution homework help

The multinomial distribution is a common way for calculating probabilities when you know that each trial has two possible outcomes, e.g., if you have a coin which comes up heads or tails with equal probability and you flip it again and again then P(X=1)=P(H)×P(T)/(P(H)+P(T)) where H and T represent two symbols for heads and tails respectively (see figure below). This can be used to calculate many probabilities starting from the same hypothesis

Confidence interval homework

A confidence interval is an interval estimate calculated from a sample data in which you have a given level of confidence, e.g., 90%, 95% or 99%, that the true value within this threshold will fall into the interval.

Hypergeometric Distribution Function

This function is used when you have a population with a finite number of individuals. In particular, it allows us to calculate how many people will exhibit one or more characteristics in this pool.

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