Cdf Interval 22D19C

1. **State the problem:** We are given the inequality $$-6 \leq x \leq -3$$ and asked to find what $F_X$ equals. 2. **Understanding $F_X$:** Typically, $F_X$ denotes the cumulative distribution function (CDF) of a random variable $X$. The CDF $F_X(x)$ is defined as $F_X(x) = P(X \leq x)$. 3. **Interpreting the inequality:** The inequality $$-6 \leq x \leq -3$$ describes the range of $x$ values. 4. **If $F_X$ is the CDF, then:** - For $x < -6$, $F_X(x) = 0$ (assuming $X$ cannot be less than $-6$). - For $-6 \leq x \leq -3$, $F_X(x)$ increases from 0 to some value. - For $x > -3$, $F_X(x) = 1$ (assuming $X$ cannot be greater than $-3$). 5. **Without additional information about the distribution of $X$,** we cannot determine the exact values of $F_X(x)$ inside the interval. However, if $X$ is uniformly distributed on $[-6, -3]$, then: $$F_X(x) = \frac{x - (-6)}{-3 - (-6)} = \frac{x + 6}{3}$$ for $$-6 \leq x \leq -3$$. 6. **Summary:** $$F_X(x) = \begin{cases} 0 & x < -6 \\ \frac{x + 6}{3} & -6 \leq x \leq -3 \\ 1 & x > -3 \end{cases}$$ This is the CDF for a uniform distribution on $[-6, -3]$.