Should the definition of random vector be associated with P: the probability set function?

The concept of random vectors is a multidimensional generalization of the concept of random variable.

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Just from $13/Page Suppose that we conduct a probabilistic experiment and that the possible outcomes of the experiment are described by sample space “\Omega” . A random vector is a vector whose value depends on the outcome of the experiment, as stated by the following definition. Let “\Omega” be a sample space. A random vector X is a function from sample space Omega to the set of K-dimensional real vectors “\mathbb{R}^{\kappa}” : “X = \Omega u2192 \mathbb{R}^{\kappa}” In rigorous probability theory, the function X is also required to be measurable. We report here a more rigorous definition of random vector by using the formulation of measure theory. Let “(\Omega, F, P)” be a probability space. Let “\beta (\mathbb{R}^{\kappa})” be the Borel sigma- algebra of “\mathbb{R}^{\kappa }” . A function “X : \Omega \rightarrow \mathbb{R}^{\kappa }” such that “\left \{ \infty \epsilon \Omega :X(\infty ) \epsilon B\right \} \epsilon F” for any “B \epsilon \beta (\mathbb{R}^{\kappa })” is said to be a random variable on “\Omega” . This definition ensures that the probability that realization of the random vector X will belong to set “B \epsilon \beta (\mathbb{R}^{\kappa})” can be defined as “P( X \epsilon B) = P({ \infty \epsilon \Omega : X(\infty ) \epsilon B})” because the set “{ \infty \epsilon \Omega : X(\infty ) \epsilon B}” belongs to the sigma- algebra F and as a consequence it’s probability is well defined. # What Will You Get? We provide professional writing services to help you score straight A’s by submitting custom written assignments that mirror your guidelines. #### Premium Quality Get result-oriented writing and never worry about grades anymore. We follow the highest quality standards to make sure that you get perfect assignments. #### Experienced Writers Our writers have experience in dealing with papers of every educational level. You can surely rely on the expertise of our qualified professionals. #### On-Time Delivery Your deadline is our threshold for success and we take it very seriously. We make sure you receive your papers before your predefined time. #### 24/7 Customer Support Someone from our customer support team is always here to respond to your questions. So, hit us up if you have got any ambiguity or concern. #### Complete Confidentiality Sit back and relax while we help you out with writing your papers. We have an ultimate policy for keeping your personal and order-related details a secret. #### Authentic Sources We assure you that your document will be thoroughly checked for plagiarism and grammatical errors as we use highly authentic and licit sources. #### Moneyback Guarantee Still reluctant about placing an order? Our 100% Moneyback Guarantee backs you up on rare occasions where you aren’t satisfied with the writing. #### Order Tracking You don’t have to wait for an update for hours; you can track the progress of your order any time you want. We share the status after each step. #### Areas of Expertise Although you can leverage our expertise for any writing task, we have a knack for creating flawless papers for the following document types. # Trusted Partner of 9650+ Students for Writing From brainstorming your paper's outline to perfecting its grammar, we perform every step carefully to make your paper worthy of A grade. ##### Preferred Writer Hire your preferred writer anytime. Simply specify if you want your preferred expert to write your paper and we’ll make that happen. ##### Grammar Check Report Get an elaborate and authentic grammar check report with your work to have the grammar goodness sealed in your document. ##### One Page Summary You can purchase this feature if you want our writers to sum up your paper in the form of a concise and well-articulated summary. ##### Plagiarism Report You don’t have to worry about plagiarism anymore. Get a plagiarism report to certify the uniqueness of your work. ## Free Features$66FREE

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