Math Problem Statement
The CDF of a random variable XX is F_X(x) = \begin{cases} 1 - e^{-2x} & x \geq 0 \ 0 & \text{otherwise} \end{cases}FX(x)={1−e−2x0x≥0otherwise find P(-4 < X < 6)P(−4<X<6).
Solution
We are given the cumulative distribution function (CDF) of a random variable :
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Cumulative Distribution Function (CDF)
Exponential Distribution
Formulas
P(a < X < b) = F_X(b) - F_X(a)
F_X(x) = 1 - e^{-2x} for x >= 0
Theorems
CDF of Exponential Distribution
Probability Rule using CDF
Suitable Grade Level
Undergraduate Level - Probability and Statistics
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