The 25th percentile is the value of $x$ for which $P(X \leq x)$ is greater than or equal to 0.25 and $P(X \geq x)$ is greater than or equal to 0.75.

First, we need to find K. Since the probabilities must sum to 1: $`$0.15 + 0.25 + 0.35 + K = 1$$ $$0.75` + K = 1$$K = 0.25$$

Now we calculate cumulative probabilities:

• $P(X \leq 1) = 0.15$
• $P(X \leq 2) = 0.15 + 0.25 = 0.40$
• $P(X \leq 3) = 0.15 + 0.25 + 0.35 = 0.75$
• $P(X \leq 4) = 1$

For the 25th percentile:

• $P(X \leq 2) = 0.40 \geq 0.25$
• $P(X \geq 2) = 0.25 + 0.35 + 0.25 = 0.85 \geq 0.75$

Both conditions are satisfied for $x = 2$, so the 25th percentile is 2.0.