We know that "X \\sim U(980,1030)" with probability density( "a = 980" and "b = 1030")
Let's find the probability density as
"{F_X}(x) = P(X < x) = \\int\\limits_{ - \\infty }^x {{p_X}(\\xi )d\\xi }"At the interval "x \\in ( - \\infty ,a]" we get "{F_X}(x) = \\int\\limits_{ - \\infty }^x {0d\\xi } = 0"
At the interval "x \\in [a,b]" we get "{F_X}(x) = \\int\\limits_a^x {\\frac{1}{{b - a}}d\\xi } = \\left. {\\frac{\\xi }{{b - a}}} \\right|_a^x = \\frac{{x - a}}{{b - a}}"
At the interval "x \\in [b, + \\infty )" we get "{F_X}(x) = \\frac{{b - a}}{{b - a}} + \\int\\limits_b^{ + \\infty } {0d\\xi } = 1"
Thus we get
a) We asked to find "P(X < 1000)"
b) We asked to find "P(X < w)" if "w \\in (980,1030)"
c) Let "Y" be the number of packages that weight less than "w" , because the events are independent "Y" is distributed according to the Binomial distribution "Y \\sim B(n,p)" with "n = 5" and "p = P(X < w) = \\frac{{w - 980}}{{50}}". The probability mass function for binomial distribtion is
where "C_n^k = \\frac{{n!}}{{k!(n - k)!}}" - binomial coefficient. Thus, the probability that all 5 net will weight less than "w"
Now we need to find the probability density for the heaviest package if we know that all 5 packages are weight less than "w" . Actually, at the previous step we found the conditional probability
Using is we can easily found it
(of course, for "980 < w < 1030" )
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