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Section 1.3-1.4

# Section 1.3-1.4 - Section 1.3 Describing Distributions...

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Section 1.3 Describing Distributions Histograms are constructed from sample data (on variable x), which may be smoothened to obtain the distribution of x over the population. Histograms of continuous data: The following represents the histogram of 100 varables of a continuous variable x. It is possible sometimes to express f(x) which describes (graph) the approximating curves. Such functions are called density function (continuous case) or distribution functions (discrete case). 1

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Continuous Distributions Definition: A function ) ( x f is called the density of a continuous variable x if (i) ; 0 ) ( x f (ii) 1 ) ( = - dx x f (a) The graph of ) ( x f is called “density curve”. (b) Proportion of x - values between ‘a’ and ‘b’ = ) ( b a dx x f = area of density curve between a and b (c ) Proportion of x - values with b x a = Proportion of x with b x a < < . ( Since the area of the curve under single value = 0). Example 1 : Let x denotes the amount of time ( hr ) the music is played in a program of a radio station. A potential sponsor wants to study the distribution of x. Suppose, the density function is 2
) ( x f = 90 x 8 (1- x ), 0 ≤ x 1 0, otherwise. (i) Note ) ( x f is a density. (ii) Proportion of programs between 0.7 and 0.9 (hr) is . 587 . 0 ) ( 9 . 7 . = dx x f (iii) The constant c such that .5 ) ( ) ( 1 = = c c O dx x f dx x f .5 10 9 90 10 9 = - c c Solving by a numerical procedure, c 2245 .838 50% of programs have music < .838 hrs.

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Section 1.3-1.4 - Section 1.3 Describing Distributions...

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