My English lesson example O The speed of cars detected by a speed camera on a section of the A1 is normally distributed with mean = 135 km/h and mean square deviation of 9 km/h. What is the probability that any car will exceed the speed limit v=130 km/h? p(130 V 135) = *(0.56) y p(V > 135) = 0.5 0.56 0 (V = 135) (V = 135) z Let us consider the probability that the speed is greater than 130 km/h as the sum of the two areas shown in colour: the first representing the probability that the speed is greater than the mean and the second corresponding to the probability that the speed is less than the mean but greater than 130 km/h. The value v = 130 corresponds to the standardized value v 130 135 z = _ = _ 0,56 9 We observe that the area in the interval [ 0.56 ; 0], due to the symmetry of the normal, is equal to that in the interval [0 ; 0.56]. Therefore: P(v > 130) = p( 0.56 z 0) + p(z > 0) = = + § + 0.5 = 0.7123 = 71.23% EXERCISES FILL IN THE GAPS 1. A probability distribution relating to a continuous random variable, of mean M(X) = .......... and mean square deviation (X) = .........., is called a normal ......................... if its trend is expressed by the function (X )2 _______ 2 2 1 ___ f(X) = _______ e 2 2. The area of the surface in colour represents the ......................... that the ......................... f(x) variable will take on a ......................... between .......... and .......... x1 x x2 3. From a ......................... variable X it is possible to move to a .................................. variable z by performing a ............................... 1 of equation X = X .......... followed by a ......................... of ratio _. 4. A random variable, once .................................., has a .................................. 1 _ through the .................................. of the function y = _ e 2 2 distribution if its .................................. is described z ___ 2 TEST 5. The graph of the density function of a standardized normal distribution: 1_ a. intersects the y-axis at _ ;0 ( 2 ) b. is symmetrical with respect to the x-axis. c. for z > 0 is decreasing; for z < 0 is increasing. 1_ d. has its maximum at 0 ; _ . ( 2 ) + 1_ z2/2 _ 6. dz = 0 e 2 468 T F T F T F T F T F