Let "X" be the time needed to fix a furnace and "X \\sim U(1.5,4)" with probability density ("b=4" and "a=1.5" )
Then let's find the distribution function as "{F_X}(x) = \\int\\limits_{ - \\infty }^x {{p_X}(\\xi )d\\xi }"
At the interval "( - \\infty ,a)" we have "{F_X}(x) = \\int\\limits_{ - \\infty }^x {0 \\cdot d\\xi } = 0"
At the interval "(a,b)" we have "{F_X}(x) = \\int\\limits_a^x {\\frac{1}{{b - a}} \\cdot d\\xi } = \\left. {\\frac{\\xi }{{b - a}}} \\right|_a^x = \\frac{{x - a}}{{b - a}}"
At the interval "(b, + \\infty )" we have "{F_X}(x) = {\\left. {\\frac{{x - a}}{{b - a}}} \\right|_{x = b}} + \\int\\limits_x^{ + \\infty } {0 \\cdot d\\xi } = 1"
Thus, it has the form
(We can use strict or non strict inequalities, it does not matter because "X" is continuous random variable).
We were asked to find such value "{q_{0.3}}" that "F({q_{0.3}}) = 0.3". We have
Thus
"{q_{0.3}} - 1.5 = 0.75"and the final answer is
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