1. **State the problem:** We want to transform random numbers uniformly distributed on the interval $[0,1]$ into random numbers uniformly distributed on the interval $[-148,1128]$. 2. **Formula used:** If $X$ is uniform on $[0,1]$, then to transform $X$ to a uniform random variable $Y$ on $[a,b]$, use the linear transformation: $$Y = a + (b - a)X$$ 3. **Apply the formula:** Here, $a = -148$ and $b = 1128$. So, $$Y = -148 + (1128 - (-148))X = -148 + (1128 + 148)X = -148 + 1276X$$ 4. **Explanation:** This formula works because when $X=0$, $Y=a=-148$, and when $X=1$, $Y=b=1128$. For values of $X$ between 0 and 1, $Y$ is linearly scaled and shifted to lie between $-148$ and $1128$. 5. **Final answer:** The transformation is $$Y = -148 + 1276X$$