Math Problem Statement
Solution
The image shows a probability density function (PDF) of a random variable . The function is defined as:
\begin{cases} kx, & 0 \leq x \leq 5 \\ k(x-3), & 5 \leq x \leq 10 \\ 0, & \text{otherwise} \end{cases}$$ The questions associated with this function are: 1. (a) Find the value of $$k$$. 2. (b) Find the expected value $$E(X)$$. 3. (c) Find the variance $$\text{Var}(X)$$. 4. (d) Find $$P(4 \leq X \leq 6)$$. 5. (e) Sketch the graph of $$f(x)$$. Let's solve each part step-by-step: ### (a) Find the value of $$k$$ To find $$k$$, we use the fact that the total area under the PDF must be 1: $$\int_{-\infty}^{\infty} f(x) \, dx = 1$$ Since the function is only nonzero from $$0$$ to $$10$$, we split the integral into two parts: $$\int_0^5 kx \, dx + \int_5^{10} k(x - 3) \, dx = 1$$ Calculating these integrals: 1. For the first part: $$\int_0^5 kx \, dx = k \left[\frac{x^2}{2}\right]_0^5 = k \cdot \frac{25}{2} = \frac{25k}{2}$$ 2. For the second part: $$\int_5^{10} k(x - 3) \, dx = k \left[\frac{(x - 3)^2}{2}\right]_5^{10} = k \cdot \frac{49}{2} = \frac{49k}{2}$$ Adding both integrals: $$\frac{25k}{2} + \frac{49k}{2} = \frac{74k}{2} = 37k$$ Setting this equal to 1: $$37k = 1 \implies k = \frac{1}{37}$$ ### (b) Find the expected value $$E(X)$$ The expected value $$E(X)$$ is given by: $$E(X) = \int_{-\infty}^{\infty} x f(x) \, dx = \int_0^5 x(kx) \, dx + \int_5^{10} x k(x - 3) \, dx$$ Substituting $$k = \frac{1}{37}$$ and evaluating: 1. For the first part: $$\int_0^5 x \cdot \frac{x}{37} \, dx = \frac{1}{37} \int_0^5 x^2 \, dx = \frac{1}{37} \cdot \frac{x^3}{3} \Bigg|_0^5 = \frac{125}{111}$$ 2. For the second part: $$\int_5^{10} x \cdot k(x - 3) \, dx = \frac{1}{37}$$Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Density Function
Expected Value
Variance
Continuous Random Variables
Formulas
Total area under the PDF must equal 1: ∫f(x)dx = 1
Expected value: E(X) = ∫x f(x) dx
Variance: Var(X) = E(X^2) - (E(X))^2
Probability for continuous random variables: P(a ≤ X ≤ b) = ∫[a, b] f(x) dx
Theorems
Basic properties of probability density functions
Suitable Grade Level
Undergraduate Level (College)
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