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Question
Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function
`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`
Find the distribution function
Solution
`f(x) = int_-oo^x f(u) "d"u`
Case 1:
x < 200
`f(x) = int_-oo^x f(u) "d"u` = 0
Case 2:
200 ≤ x ≤ 600
`f(x) = int_-oo^x f(u) "d"u`
= `int_-oo^200 f(u) "d"u + int_200^x f(u) "d"u`
= `0 + k int_200^x "d"u`
= `1/400 [u]_200^x`
= `1/400 (x - 200)`
= `x/400 - 1/2`
Case 3:
x > 600
`f(x) = int_oo^x f(u) "d"u`
= `int_-oo^200 f(u) "d"u + int_200^600 f(u) "d"u + int_600^x f(u) "d"u`
= `0 + k int_200^600 "d"u + 0`
= `1/400 [u]_200^600`
= `400/40`
= 1
`f(x) = {{:(0",", "for" x < 200), (x/400 - 1/2",", "for" 200 ≤x ≤ 600),(1",", "for" x > 600):}`
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