The normal law of distribution of a random variable with the parameters a = 0, s 2 = 1, i.e. The mathematical expectation of a random variable X distributed under the normal law is M(X) = a, and its dispersion D(X) = s 2. Some examples of this behavior are the height of a man, the velocity in any direction of a molecule in gas, and the error made in measuring a physical quantity.Ī continuous random variable X has the normal law of distribution (Gauss law) with the parameters a and s 2 if its probability density has the following form:
The central limit theorem, one of the two most important results in probability theory, gives a theoretical base to the often noted empirical observation that, in practice, many random phenomena obey, at least approximately, a normal probability distribution. This result was later extended by Laplace and others and is now encompassed in a probability theory known as the central limit theorem, which is discussed later. The normal distribution was introduced by the French mathematician Abraham DeMoivre in 1733 and was used by him to approximate probabilities associated with binomial random variables when the binomial parameter n is large. The main feature allocating it among other laws is that it is the limiting law to which other laws of distribution are approximated at rather frequently meeting typical conditions. The normal law of distribution is most frequently met in practice. The required probability P( X ³ 20) can be found by integrating the probability density:īut it is simply to do this by using the distribution function:įind the mean square deviation: s(X) = M(X) = 15 days. Solution: By the hypothesis the mathematical expectation M(X) = 1/l = 15, and consequently l = 1/15, and the probability density and the distribution function have the following form: Find the probability density, the distribution function and the mean square deviation of the random variable X.
#CDF EXPONENTIAL DISTRIBUTION TV#
Determine the probability that it is required no less than 20 days on repair of a TV if the average time of repair of TVs makes 15 days. It has been established that a time of repair of TVs is a random variable X distributed under the exponential law. The exponential law of distribution plays the big role in the theory of mass service and the theory of reliability.Įxample. Its mathematical expectation and its dispersionįrom the theorem follows that for a random variable distributed under the exponential law, the mathematical expectation is equal to the mean square deviation, i.e. The function of distribution of a random variable X distributed under the exponential law is The distribution curve j(x) and the graph of function of distribution F(x) of the random variable X are the following: For instance, the amount of time (starting from now) until an earthquake occurs, or until a new war breaks out, or until a telephone call you receive turns out to be a wrong number are all random variables that tend in practice to have exponential distributions. The exponential distribution often arises, in practice, as being the distribution of the amount of time until some specific event occurs.
A continuous random variable X has the exponential law of distribution with the parameter l > 0 if its probability density has the following form: