Random Max

AtCoder

IDhhkb2020_f

Time3000ms

Memory256MB

Difficulty—

There are $N$ continuous random variables $x_i$ ($1 ≤ i ≤ N$), each of which follows the continuous uniform distribution on the interval $[L_i, R_i]$. (That is, $x_i$ is a random variable that can take any real value between $L_i$ and $R_i$ (inclusive) with equal probability.) Let $E$ be the expected value of the maximum of these $N$ random variables. Under the constraints in this problem, it can be proved that $E \times (N+1)! \times \prod_{i=1}^N (R_i - L_i)$ is a positive integer. Find this value modulo $1,000,000,007$. ## Constraints * $1 ≤ N ≤ 1000$ * $0 ≤ L_i < R_i ≤ 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $L_1$ $R_1$ $:$ $L_N$ $R_N$ [samples]

Samples

Input #1

1
1 2

Output #1

3

The expected value of the maximum of these random variables - there is actually just one in this case - is equal to the median of the range of the values the variable can take, that is, $E = \frac{3}{2}$.
Thus, the correct output is $E \times (N+1)! \times (R_1 - L_1) = E \times 2 = 3$.

Input #2

2
1 2
1 2

Output #2

10

The expected value in question is $E = \frac{5}{3}$.

Input #3

2
1 2
2 4

Output #3

Input #4

Output #4

52776507

API Response (JSON)

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  "problem": {
    "name": "Random Max",
    "description": {
      "content": "There are $N$ continuous random variables $x_i$ ($1 ≤ i ≤ N$), each of which follows the continuous uniform distribution on the interval $[L_i, R_i]$. (That is, $x_i$ is a random variable that can tak",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "hhkb2020_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ continuous random variables $x_i$ ($1 ≤ i ≤ N$), each of which follows the continuous uniform distribution on the interval $[L_i, R_i]$. (That is, $x_i$ is a random variable that can tak...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments