My English lesson probability Simulation of 25 rolls of a die. Expected event: value 1 or 2 ( p = 1/3) 0,2 0,15 0,1 0,05 0 0 2 4 6 8 10 12 14 16 18 number of hits 1__ = _____ e 2 1__ e = _____ 2 (X )2 _______ 2 2 ( X + ) ___________ 1 ___ f(X) = _______ e 2 2 2 (X )2 ___________ 2 2 This function: Q is defined for each x R; Q takes values that are always positive; Q is symmetrical with respect to the line X = ; Q has its own maximum at the mean value: 1 ___ f( ) = _______ 2 = f(X) f (x) Typical bell-shaped patterns are those of the distributions relating to the age classes into which a population is divided or to some constant characteristics measured in large samples. For example, we can observe that even the distribution of the discrete random variable X = «number of times a 1 or a 2 has been tossed in n consecutive rolls of an unloaded dice , takes on a bell-shaped pattern if we make a large number of rolls assuming continuity characteristics (figure beside). A probability distribution relating to a continuous random variable, with mean M(X) = and mean square deviation (X) = , is called a normal distribution if its trend is expressed by the following function: (2 X )2 ___________ 2 2 2 1__ e = _____ 2 In the 1st paragraph, we have seen that some histograms (e.g. those relating to the number of Heads on n tests or to heights) have a similar shape, which, as the classes considered increase, comes closer and closer to that of a bell-shaped curve. This important continuous probability distribution, to which the previous ones are approximated, is called the normal distribution. DEFINITION BE CAREFUL! B f(2 f( f 2 X) = The normal distribution p(x1 X x2) The curve represents the density function of a continuous random variable: the probability that this variable takes on a value belonging to the interval [x1 ; x2] is equal to the area of the region enclosed by the curve, by the x-axis and by the x2 straight-lines of equation X = x1 and X = x2 that is p(x 1 X x 2) = x1 x2 x2 x BE CAREFUL! B Th random variable X = X The is in fact the variable «deviation from the mean . This variable has mean 0 since, as we know, the arithmetic mean of the deviations from the mean is precisely 0. BE CAREFUL! B A standardised variable measures deviations from the mean in sigma units and, since variable X and its have the same unit of measurement, the relationship is independent of the unit of measurement used. 466 f(X)dX x1 Right because the curve represents the trend of the density function of a random variable, the value of the entire area subtended by this curve is equal to 1. Variations in the characteristic shape of the curve, the mean value being equal, depend essentially on the value of the mean square deviation; the larger it is, the greater the dispersion; this fact, in the representation of a continuous distribution, is evidenced by a greater «openness of the bell. 1 Then, by performing a translation of equation X = X and a strain of ratio _, we obtain a new variable, usually denoted z, which is called the standardised variable. f(x) M(X ) = 0 M(X) = x From the random variable X, which has the mean value and the mean square deYLDWLRQ ZH WKXV PRYH RQ WR WKH FRUUHVSRQGLQJ YDULDEOH LQ standardised form: X z=_ which has its mean value equal to 0, variance equal to 1 and, of course, also mean square deviation equal to 1.