Handout_8 THE NORMAL PROBABILITY DISTRIBUTION

Handout_8 THE NORMAL PROBABILITY DISTRIBUTION - Apr 709 THE...

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Apr 7’09 THE NORMAL PROBABILITY DISTRIBUTION #8 FOR A CONTINUOUS RANDOM VARIABLE A random variable is a variable whose numerical value depends on chance factors and cannot be exactly predicted in advance. There are two types of random variables: discrete and continuous . A discrete variable is characterized by gaps or interruptions in the values that it can take. A number of heads observed in an experiment of tossing simultaneously 10 coins may serve as an example of a discrete random variable which may take on the whole values from 0 to 10. A continuous random variable can take ANY value within a specified relevant interval of values; the possible values of such a variable have no any gaps or interruptions . Examples : height, weight, temperature, or ages of a patient. (Usually the rounded off results of measurements of these quantities are recorded. They look like discrete numbers, but principally, with a perfect measuring device, these quantities can be measured with any desired accuracy). A probability distribution of a discrete random variable x is a rule that associates each possible value x of this variable with the probability P(x) of occurring of this value in a chance experiment. The examples of discrete probability distributions are the binomial and Poisson distributions (see handouts ##6a, 7). Similarly, it is possible to introduce a probability density function f(x) which characterizes a chance experiment in the case of a continuous random variable. By definition , a nonnegative function f(x) is called a probability density function ( pdf, sometimes called probability distribution ) of the continuous random variable X if a total area bounded by this curve and the x-axis is equal to 1 and if a subarea under the curve bounded by the curve, the x-axis, and perpendiculars erected at any two points a and b gives the probability P(a ≤ X ≤ b) that a randomly selected score X falls in the interval [a, b] (see Fig. 2 below). The following example can help us to understand the sense of a probability density function of the continuous random variable. Suppose a bank is interested in improving its services to customers. The manager decides to begin with finding the amount of time tellers spend on each transaction, rounded off to nearest minute. The times for n = 75 different transactions have been recorded, then grouped in the classes of 1-min length, and the relative frequency histogram with a corresponding frequency polygon has been drawn (Fig. 1a). The heights of the bars represent the relative frequencies which have a sense of empirical probabilities. Fig. 1a Fig. 1b 1
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Let us increase the number n of transactions under the observation, and let us increase the precision with which the time of each transaction is measured. Theoretically at least, the transactions could be timed to the nearest tenth of a minute, or hundredth of a minute, or even more precisely. In each case, a histogram and frequency polygon could be drawn. If
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