The CDF of a random variable X*X* is
F_X(x) = \begin{cases} 1 - e^{-2x} & x \geq 0 \\ 0 & \text{otherwise} \end{cases}*FX*​(*x*)={1−*e*−2*x*0​*x*≥0otherwise​
find P(-4 < X < 6)*P*(−4<*X*<6).

We are given the cumulative distribution function (CDF) \( F_X(x) \) of a random variable \( X \):

\[
F_X(x) = 
\begin{cases} 
1 - e^{-2x} & x \geq 0 \\
0 & \text{otherwise}
\end{cases}
\]

We are tasked with finding \( P(-4 < X < 6) \), which represents the probability that the random variable \( X \) lies between \(-4\) and \(6\).

### Step 1: Recognize the relation between CDF and PDF
The CDF \( F_X(x) \) gives us the probability that \( X \) is less than or equal to a given value \( x \), i.e., \( F_X(x) = P(X \leq x) \). The probability we seek can be computed as:

\[
P(-4 < X < 6) = F_X(6) - F_X(-4)
\]

### Step 2: Evaluate \( F_X(x) \) at 6 and -4
- For \( x = 6 \), since \( 6 \geq 0 \), we use the expression \( 1 - e^{-2x} \):
  \[
  F_X(6) = 1 - e^{-2 \cdot 6} = 1 - e^{-12}
  \]

- For \( x = -4 \), since \( -4 < 0 \), we use the CDF definition for negative \( x \), which is 0:
  \[
  F_X(-4) = 0
  \]

### Step 3: Compute the probability
Now, we compute \( P(-4 < X < 6) \) as:

\[
P(-4 < X < 6) = F_X(6) - F_X(-4) = (1 - e^{-12}) - 0 = 1 - e^{-12}
\]

### Final Answer:
\[
P(-4 < X < 6) = 1 - e^{-12}
\]

Let me know if you'd like further details on any part of this solution or any clarification.

Here are some related questions for you:
1. What is the PDF corresponding to the given CDF?
2. How would you calculate \( P(X > 10) \) using the CDF?
3. How does the CDF change if the rate parameter changes from 2 to 3?
4. Can you verify the result using integration of the PDF between \(-4\) and \(6\)?
5. What is the mean and variance of \( X \), assuming it follows an exponential distribution?

**Tip:** The CDF of a random variable provides an easy way to compute probabilities for ranges without having to integrate the probability density function each time.

The CDF of a random variable X is F_X(x) = \begin{cases} 1 - e^{-2x} & x \geq 0 \\ 0 & \text{otherwise} \end{cases}. Find P(-4 < X < 6).

Learn how to find the probability P(-4 < X < 6) for a random variable X with a given cumulative distribution function (CDF). This problem involves using key concepts of probability theory, including CDF and exponential distribution, with step-by-step instructions suitable for undergraduate students.

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