by **HallsofIvy** » Tue Jan 12, 2021 4:26 pm

The first one says that "the probability that the random variable, X, is greater than or equal to [tex]x_c[/tex] and less than or equal to [tex]x_d[/tex] is equal to the sum, from i= c to i= d of [tex]f(x_i)[/tex] and that is equal to the sum, from i= c to i= d of [tex]\frac{1}{n}[/tex]". That last equality is a bit peculiar. It says that [tex]f(x_i)= \frac{1}{n}[/tex], a constant, for all i. Because of that all three sums are [tex]\frac{1}{n}(d- c)[/tex].

The second says that "The probability that the random variable, x, is greater than or equal to c and larger than or equal to d is the integral, from x= c to x= d of f(x) with respect to x and that is equal to the integral, from c to d, of [tex]frac{1}{b- a} dx[/tex] which is equal to the fraction [tex]\frac{d- c}{b- a}[/tex]. That is, the function, f(x), is the constant function, [tex]f(x)=\frac{1}{b-a}[/tex]". Of course, because [tex]\frac{1}{b- a}[/tex] is a constant, the integral is [tex]\int_c^d \frac{1}{b- a}dx= \frac{1}{b- a}\int_c^d dx= \frac{1}{b-a}(d- c)= \frac{d- c}{b- a}[/tex].

The third says that "the function f(x) is equal to the constant [tex]\frac{1}{b- a}[/tex] as long as x is larger than or equal to a and less than or equal to b and equal to 0 for any other x (less than a or larger than b)."