Let $N{}$ be a fixed positive integer, $S{}$ be the set $\{1, 2,\ldots , N\}$ and $\mathcal{F}$ be the set of functions $f:S\to S$ such that $f(i)\geqslant i$ for all $i\in S.$ For each $f\in\mathcal{F}$ let $P_f$ be the unique polynomial of degree less than $N{}$ satisfying $P_f(i) = f(i)$ for all $i\in S.$ If $f{}$ is chosen uniformly at random from $\mathcal{F}$ determine the expected value of $P_f'(0)$ where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\][i]Proposed by Ishan Nath[/i]

1. **Define the problem and the given sets:**
   - Let \( N \) be a fixed positive integer.
   - Let \( S \) be the set \( \{1, 2, \ldots, N\} \).
   - Let \( \mathcal{F} \) be the set of functions \( f: S \to S \) such that \( f(i) \geq i \) for all \( i \in S \).

2. **Define the polynomial \( P_f \):**
   - For each \( f \in \mathcal{F} \), let \( P_f \) be the unique polynomial of degree less than \( N \) satisfying \( P_f(i) = f(i) \) for all \( i \in S \).

3. **Objective:**
   - Determine the expected value of \( P_f'(0) \) where \( P_f'(0) = \frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0} \).

4. **Consider the linearity of the interpolation polynomial:**
   - For \( x_1, \ldots, x_N \in \mathbb{R} \), let \( P(x, x_1, \ldots, x_N) \) be the unique polynomial of degree less than \( N \) satisfying \( P(i) = x_i \) for \( i \in S \).
   - The map \( \mathbb{R}^N \to \mathbb{R}[x] \) given by \( (x_1, \ldots, x_N) \to P(x, x_1, \ldots, x_N) \) is linear.
   - The map \( \mathbb{R}[x] \to \mathbb{R} \) given by \( P(x) \to P'(0) \) is linear as well.

5. **Use the linearity of expected value:**
   - By the linearity of expected value, we have:
     \[
     \mathbb{E}(P'(0, f(1), \ldots, f(N))) = P'(0, \mathbb{E}(f(1)), \ldots, \mathbb{E}(f(N))).
     \]

6. **Determine the expected values \( \mathbb{E}(f(i)) \):**
   - Since \( f(i) \geq i \) and \( f(i) \) is chosen uniformly from \( \{i, i+1, \ldots, N\} \), the expected value of \( f(i) \) is:
     \[
     \mathbb{E}(f(i)) = \frac{i + (i+1) + \cdots + N}{N - i + 1} = \frac{(N - i + 1)(i + N)/2}{N - i + 1} = \frac{i + N}{2}.
     \]

7. **Construct the polynomial with expected values:**
   - The polynomial \( P(x, \mathbb{E}(f(1)), \ldots, \mathbb{E}(f(N))) \) is:
     \[
     P(x, \frac{1+N}{2}, \frac{2+N}{2}, \ldots, \frac{N+N}{2}).
     \]

8. **Determine the derivative at \( x = 0 \):**
   - Since the polynomial \( P(x) \) is linear in the expected values, the derivative \( P'(0) \) will be the same as the derivative of the polynomial interpolating the points \( (i, \frac{i+N}{2}) \).

9. **Calculate the derivative:**
   - The polynomial \( P(x) \) that interpolates the points \( (i, \frac{i+N}{2}) \) is a linear function of \( x \) with a slope of \( \frac{1}{2} \). Therefore, \( P'(0) = \frac{1}{2} \).

The final answer is \( \boxed{\frac{N+1}{2}} \).