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.
Not very convenient numbers but it simplifies a bit
1.5(cbrt(0.2)-cbrt(0.16))-0.02 = 0.04288
Basically an integral calculus problem to find the area of the region in the unit square bounded by the random variable at 0.16 and 0.20.
The algebra is done out here
https://www.desmos.com/calculator/amqek9lp7t
For part 2 you need to find a series. Each term is similar to the above.
Do it 8 more times for the bonus.
Here's a picture of the regions you need to sum.
Part 2 is the red regions
https://www.desmos.com/calculator/e9xvfihsdu
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Posted by Jer
on 2025-02-27 10:19:38 |
Part 1
