Question

A beginning unicyclist is attempting to unicycle from his garage to the end of his driveway—a distance of 200 feet. His skill is such that, once mounted on the unicycle, he will fall within a distance of x feet from his starting point with probability given by x/250 − x2/5002, for 0 < x < 500. Let X be the distance from the unicyclist’s starting point at which he falls.
(a) GivethecdfFX(x)ofX forallx∈R.
(b) GivethepdffX(x)ofX forallx∈R.
(c) With what probability does the unicyclist fall before reaching the end of his driveway?
(d) With what probability does the unicyclist reach the end of his driveway before falling?
(e) What is the maximum distance the unicyclist is capable of reaching?
(f) Find the median of X.
(g) Find EX.
(h) Find the standard deviation of X.
(i) Let Y = (200 − X)/3 represent the remaining distance, in yards, to the end of the driveway from the garage (if he passes the end of the driveway, Y will be a negative number).
i. Find EY .
ii. Find Var Y .

Answer 1

GIVEN THAT:-

According to the question we have that :

A beginning unicyclist is attempting to unicycle from his garage to the end of his driveway—a distance of 200 feet

now finding the solutions for the questios which are asked below:

TO FIND :-a) GivethecdfFX(x)ofX forallx∈R.

now from the above we have that

The CDF of X is

TO FIND :-b) GivethepdffX(x)ofX forallx∈R

SO according to the data we ahve that

The PDF is found by differentiating,

TO FIND :-c)  With what probability does the unicyclist fall before reaching the end of his driveway?

so now the unicyclist fall before reaching the end of his driveway the probability is

TO FIND :-d)With what probability does the unicyclist reach the end of his driveway before falling?

P(X=200) = 0

SINCE

For continuous distribution, the point probability is 0

TO FIND :e)What is the maximum distance the unicyclist is capable of reaching?

in the above data it is clear that the maximum distance the unicyclist is capable of reaching is maximum value of X=500

TO FIND :-f)Find the median of X

now for fonding the median of X we use median as m

there fore m=146.44661

it is because

TO FIND :-g)Find EX

now we need to find the value of EX as it cannot be accurate we can find the the expected value

there fore the expected value is E(X) = 166.6667

it is because of

PLEASE GIVE IS UPVOTE IT IS VERY HELP FULL TO US