Each of x and y is a real number chosen randomly from the interval (0,1).
z is a random variable defined as z = x3*y
1) Find the probability that z is in the interval (.16,.20)
2) Find the probability that z is any decimal whose first digit
when written in scientific notation is a 1.
Bonus: Find the full probability distribution of the first digit of z.
Note: Provide all answers as exact numbers in closed form.
Very similar to part 1 but in preparation to tackle part 2.
Part 1.1) Find the probability that z is in the interval (0.1,0.2)
cbrt(.2)-cbrt(.1)+[integral(from cbrt(.2) to 1).2/x^3 dx]-[integral(from cbrt(.1) to 1).1/x^3 dx]
=cbrt(.2)-cbrt(.1)-.2/2+.2/(2cbrt(.2)^2)+.1/2-.1/(2cbrt(.1))
=3cbrt(.2)/2-3cbrt(.1)/2-1/20
=(3/2)(cbrt(.2)-cbrt(.1)-1/30)
=(3/2)(cbrt(.1)(cbrt(2)-1)-1/30)
Part 1.2) Find the probability that z is between 10^-n and 2*10^-n
Similar integral to above simplifies to
(3/2)(cbrt(10^-n)(cbrt(2)-1)-10^-n/3)
Part 2)
SUM = Sum the above from 0 to infinity
separate into
(3/2)(cbrt(2)-1)SUM(cbrt(10^-n))-(1/2)SUM(10^-n)
The sums are (10^(2/3)+10^(1/3)+1)/9 and 1/9
Simplify to
3(cbrt(2)-1)(cbrt(100)+cbrt(10)+1)-1)/18 = 0.2821695477
|
|
Posted by Jer
on 2025-02-28 13:19:08 |
Part 2
