tailieunhanh - Handbook of mathematics for engineers and scienteists part 154

Handbook of mathematics for engineers and scienteists part 154. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | . Random Variables and Their Characteristics 1039 . Random Variables and Their Characteristics . One-Dimensional Random Variables . Notion of random variable. Distribution function of random variable. A random variable X is a real function X X w we Q on a probability space Q such that the set w X w x belongs to the a-algebra F of events for each real x. Any rule table function graph or otherwise that permits one to find the probabilities of events A QF is usually called the distribution law of a random variable. In general random variables can be discrete or continuous. The cumulative distribution function of a random variable X is the function Fx x whose value at each point x is equal to the probability of the event X x Fx x F x P X x . Properties of the cumulative distribution function 1. F x is bounded . 0 F x 1. 2. F x is a nondecreasing function for x e -to to . if x2 x1 then F x F x1 . 3. lim F x F -to 0. x - 4. lim F x F to 1. x 5. The probability that a random variable X lies in the interval x1 xf is equal to the increment of its cumulative distribution function on this interval . P x1 X x2 F xf - F x1 . 6. F x is left continuous . lim F x F x0 . x -xo-0 . Discrete random variables. A random variable X is said to be discrete if the set of its possible values the spectrum of the discrete random variable is at most countable. A discrete distribution is determined by a finite or countable set of probabilities P X xf such that P X xi 1. i To define a discrete random variable it is necessary to specify the values x1 xf and the corresponding probabilities p1 pf where pi P X xi . Remark. In what follows we assume that the values of a discrete random variable X are arranged in ascending order. In this case the cumulative distribution function of a discrete random variable X is the step function defined as the sum F x P X xi . xi x It is often convenient to write out the cumulative distribution function .