Random Seed Utilization

Solution & Explanation

Understanding the Problem:

We have two functions:

• Function X: Generates a random integer between a minimum value, $N$, and a maximum value, $M$.
• Function Y: Uses the output of Function X as its maximum value, with the same minimum value $N$.

Distribution of Samples from Function Y:

1. Function X Distribution:

   • $X$ is a discrete uniform random variable over $[N, M]$.
   • Probability Mass Function (PMF) of $X$: $p_X(x) = \frac{1}{M - N + 1}\quad \text{for}\quad x \in [N, M]$

2. Function Y Distribution Given $X = x$:

   • $Y$ is a discrete uniform random variable over $[N, x]$.
   • Conditional PMF of $Y$ given $X = x$: $p_{Y|X}(y|x) = \frac{1}{x - N + 1}\quad \text{for}\quad y \in [N, x]$

3. Joint Distribution of $X$ and $Y$:

   • $p_{X,Y}(x,y) = p_X(x) \cdot p_{Y|X}(y|x)$
   • $p_{X,Y}(x,y) = \frac{1}{M - N + 1} \cdot \frac{1}{x - N + 1}$

4. Marginal Distribution of $Y$:

   • To find the marginal distribution $p_Y(y)$, we integrate over all possible values of $X$: $p_Y(y) = \sum_{x=y}^{M} p_{X,Y}(x,y) = \sum_{x=y}^{M} \frac{1}{(M-N+1)(x-N+1)}$
   • This results in a non-uniform distribution for $Y$, as the probability of $Y$ decreases as $y$ approaches $M$.

Expected Value of Samples from Function Y:

1. Using the Law of Iterated Expectation:

   • $E[Y] = E[E[Y|X]]$

2. Expected Value of $Y$ Given $X = x$:

   • $E[Y|X=x] = \frac{N + x}{2}$

3. Overall Expected Value of $Y$:

   • $E[Y] = E\left[\frac{N + X}{2}\right] = \frac{1}{2} \left(N + E[X]\right)$
   • $E[X] = \frac{M + N}{2}$ (since $X$ is uniform over $[N, M]$)
   • $E[Y] = \frac{1}{2} \left(N + \frac{M + N}{2}\right) = \frac{3N + M}{4}$

Conclusion:

• Distribution of $Y$: Non-uniform, with probabilities decreasing as $y$ approaches $M$.
• Expected Value of $Y$: $\frac{3N + M}{4}$, which reflects the influence of both the uniform distribution of $X$ and the conditional constraints on $Y$.

The problem illustrates how conditional distributions and expectations can be derived from basic principles of probability and statistics, demonstrating the non-uniform nature of $Y$ despite $X$ being uniformly distributed.