12341. Square of fortune

A real number x is chosen uniformly at random from the segment [0; n]. The player’s reward equals the square of the chosen number: W = x².

Compute the expected value of the reward.

Input. One real number n (0 ≤ n ≤ 10⁹).

Output. Print the expected value of the reward with at least 4 decimal digits.

Sample input	Sample output
10	33.333333

SOLUTION

probability

Algorithm analysis

The random variable X is uniformly distributed on the segment [0; n]. The probability density function on this interval is

f(x) =

To find the expected value of the reward E[W] = E[x²], we need to average x² over all possible values of x:

E[x²] = = = =

Example

For n = 10 the answer is

E[x²] = = 33.333333

Algorithm implementation

Read the input value n.

scanf("%lld", &n);

Compute and print the answer.

res = n * n / 3.0;

printf("%lf\n", res);