Math Problem Statement
Solution
The problem provides a probability density function (PDF) for a random variable . The function is piecewise:
\begin{cases} Kx & \text{for } 0 \leq x \leq 5 \\ K(x-3) & \text{for } 5 < x \leq 10 \\ 0 & \text{otherwise} \end{cases}$$ The questions involve: 1. **Finding the value of $$K$$** using the property that the total probability over the entire range must equal 1. 2. **Expected value $$E(X)$$**. 3. **Variance $$\text{Var}(X)$$**. 4. **Probability $$P(4 \leq X \leq 6)$$**. 5. **Sketching the graph of $$f(x)$$**. Let's solve each part: ### (a) Finding the value of $$K$$ To find $$K$$, we use the property that the total area under the PDF must equal 1: $$\int_{-\infty}^{\infty} f(x) dx = 1$$ This simplifies to calculating the area for the two intervals $$[0, 5]$$ and $$[5, 10]$$. $$\int_0^5 Kx \, dx + \int_5^{10} K(x-3) \, dx = 1$$ Let me calculate $$K$$ first.The value of $$K$$ is $$\frac{1}{35}$$. ### (b) Finding the expected value $$E(X)$$ The expected value $$E(X)$$ is given by: $$E(X) = \int_{-\infty}^{\infty} x f(x) dx$$ This will involve two integrals over the intervals $$[0, 5]$$ and $$[5, 10]$$: $$E(X) = \int_0^5 x \cdot \frac{1}{35} x \, dx + \int_5^{10} x \cdot \frac{1}{35} (x-3) \, dx$$ Let me compute this next.It seems like I can’t do more advanced data analysis right now. Please try again later. However, here’s the approach to compute the expected value: 1. Set up the integral for $$E(X)$$: $$E(X) = \int_0^5 x \cdot \frac{1}{35} x \, dx + \int_5^{10} x \cdot \frac{1}{35} (x-3) \, dx$$ 2. Calculate each integral for the two regions and sum them to get $$E(X)$$. Would you like help with the next part, or have any questions about these steps? Here are some related questions: 1. How do we interpret the expected value in probability theory? 2. How is the variance $$\text{Var}(X)$$ calculated after finding $$E(X)$$? 3. How do we compute probabilities over specific intervals like $$P(4 \leq X \leq 6)$$? 4. What is the importance of normalizing the probability density function (finding $$K$$)? 5. How would the graph of $$f(x)$$ look like based on the function definition? **Tip**: When solving integrals of piecewise functions, remember to treat each piece independently over its specific interval.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Density Function
Expected Value
Variance
Piecewise Function
Formulas
Normalization condition: ∫ f(x) dx = 1
Expected Value: E(X) = ∫ x f(x) dx
Variance: Var(X) = E(X^2) - [E(X)]^2
Probability: P(a <= X <= b) = ∫_a^b f(x) dx
Theorems
Fundamental Theorem of Calculus
Basic Probability Theorem
Suitable Grade Level
Undergraduate (First Year Calculus/Statistics)
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