Law of total probability

The recursive approach uses an elementary law of conditional expectation. Let A be an event and A its complement. Let Y be a random variable, E Y) its expectation (or average value) and E Y A) is conditional expectation, given that the event A has occurred. P A) is the probability that event A occurs. Then the law of total probability for expectation is [31] ... 

The law of total probability is very useful in the context of conditional probability. If an event A is subdivided into N (either finite or countably infinite) mutually exclusive events, Ai, A2,An, then the probability of another event B is given by [177] ... 

The continuous analogy of the law of total probability is also commonly used. In this case, the probability of event B can be expressed as ... 

Eurthermore, the probability of the other two joint events can be obtained by the law of total probability ... 

On the other hand, the scores of the rest of the students (group B) also follow the log-normal distribution with mean 480 and standard deviation 15 so the parameters are xb = 6.1733 and a = 9.7609 x 10 for this group. Then, by using the law of total probability, the score distribution of all students is given by ... 

The plausibility P(Cj V, U) can be used not only for selection of the most plausible class of models, but also for response prediction based on all the model classes. Let Q denote a quantity to be predicted, e.g., first story drift. Then, the PDF of Q given the data V can be calculated from the law of total probability as follows ... 

From the posterior distribution we can deduce for example, a (1 — a). 100% credibility interval for fi and for 6. We can also compute the predictive distribution for anew measurement X, using the law of total probability ... 

Law of Total Probability n Also known as the Theorem on Total Probability or the Law of Alternatives. It states that the probability, P(A) of an event, A, is equal to the sum of the conditional probabilities of A, given events, Ej, P(A 1 ), times the probability of event, , for i = 1, 2, 3,. ..,N where Nisa. positive integer or infinity, and where Efi are non-overlapping and form a partition of a sample space that covers the sample space of A. This can be expressed as ... 

Theorem on Total Probability n An alternate name for law of total probability. 

Adequate hazard scenarios and relevant occurrence probabilities need to be estimated, often on the basis of expert assessments and judgements. When for mutually independent hazard situations H, the failure F of the component given a particular situation occurs with the conditional probability P(i Ef,), then the total probability of failure Pj is given by the law of total probability as ... 

Arguably it may be no easier to assign a well-founded probability interval than an exact probability Suppose for example that we have assigned a well-founded unconditional probability distribution F x Z, K), but have problems assigning a probability distribution of Z (to integrate with F using the law of total probability), where Z is known to take a value in the interval Then we may at least... 

Once the system model is constructed by connecting models of components, we execute the model by using an algorithm similar to FPTC. Tokens in FPTA consist of two elements a mode and its probability. The technique to deal with the computation of the modes is as same as the fix-point technique used in FPTC. The law of total probability is used to calculate the probability associated with each mode. 

In FPTA, if there are n possible modes that can be transitioned to a particular failure of an output, the law of total probability says ... 

The term p A Mi) in the numerator on the right-hand side of Eq. 9 is the evidence (sometimes also referred to as the model class likelihood) for the model class Al, provided by the data d. The evidence, hereafter denoted with e, is a very important quantity in Bayesian model class selection and can be determined based on the law of total probability as... 

If A is an event and its complement is A, and Y is a random variable, E (Y) its mathematical expectation (or average value), and E(Y A) its conditional expectation given that the A event has occurred, then the law of total probability for the expectations is written as... 

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