Uses it aids solving many business and economic

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Uses It aids solving many business and economic problems including the problems in social and physical sciences. Hence, it is cornerstone of modern statistics. It becomes basis to know how far away and in what direction variable from its population mean. It is symmetrical. Hence, mean, median and mode are identical and can be known. It has only one maximum point at the mean, and hence it is unimodel (i.e. only one mode).
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44 Definition In mathematical form, the normal probability distribution is defined by 2 2 2 ) x - x ( e - 2 N Y σ π ε Where Y = ordinate of the curve at a point x N= number of items in the frequency distribution = standard deviation of distribution = 3.14159 or (22/7) e= 2.71828 The above equation can also be written as 2 2 o 2 x - Y Y σ ε ε Where π σ 2 N Y o If the normal curve is in terms of standard deviation units, it is called normal deviate. The normal deviate at the mean will be zero viz. x = x/ = 0/ = 0, when x x and at values equal to 10, 20 and 30 will be respectively 1 , 2 and 3 (to the left of the mean, these units will be negative as x variates will be less than the value of mean, and to the right of the mean, these units will be positive as x variate will be greater than the value of mean). This is known as changing to standardized scale. In equation the charging to standardized scale is written. σ σ x x - x Z The normal curve is distributed as under : a. Mean 1 covers 68.27% area; 34.135% on either side of the mean. b. Mean 2 covers 94.45% area; 47.725% on either side of the mean. c. Mean 3 covers 99.73% area; 49.865% on either side of the mean. Method of ordinates To make a curve on graph, we need the frequencies and the values of variable which represent on the ordinate (Y axis) and abscissa (X-axis) respectively. Hence, in order to fit a curve we must know the ordinates (i.e. frequencies) at the various points of the abscissa scale. Find the x , N and class interval i, if any, of the observed distribution. Then calculate Y 0 . σ π i 0 i i 0 N 0.399 Y or 2.5071 N 2 N Y This given mean ordinate.
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45 5.4. REVISION POINTS ipi x (X) E n 1 i Variance = (x) 2 - [ (x) ] 2 Binomial distribution P(r)= nc r q n r p r Poisson distribution P(r) = r! m e r) (X p r -m 5.5 INTEXT QUESTIONS 1) What is meant by the Poisson distribution? What are its uses? 2) A systematic sample of 100 passes was taken from the concise Oxford Dictionary and the observed frequency distribution of foreign words per page was found to be as follows: page (x) 0 1 2 3 4 5 6 Frequency 48 27 12 7 4 1 1 Calculate the expected frequencies using Poisson distribution. Also calculate the Variance of fitted distribution Ans.: 37, 37, 18, 16, 2, 0, 0 Variance = 0.99 Income of a group of 10000 persons was found to be normally distributed with mean Rs.750 per month and standard deviation Rs.50. Show that of third group about 95 per cent had income exceeding Rs.668 and only 5 per cent had income exceeding Rs.832. What was the lowest income among the richest 100. Ans : Rs.866.
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