Math Problem Statement
Solution
The given problem involves finding and for a continuous random variable with the probability density function (pdf):
2e^{-2x}, & x > 0 \\ 0, & \text{otherwise} \end{cases}$$ ### Step 1: Compute $$E[X^2]$$ The expected value $$E[X^2]$$ is given by: $$E[X^2] = \int_{-\infty}^{\infty} x^2 f(x) \, dx$$ Since the pdf is non-zero only for $$x > 0$$, we only integrate from 0 to infinity: $$E[X^2] = \int_0^{\infty} x^2 \cdot 2e^{-2x} \, dx$$ To solve this, we can use integration by parts or recognize it as a gamma function integral. The result of this standard integral is: $$E[X^2] = \frac{3}{2}$$ ### Step 2: Compute $$E[X]$$ To find the variance, we also need $$E[X]$$, the expected value of $$X$$: $$E[X] = \int_{0}^{\infty} x \cdot 2e^{-2x} \, dx$$ Again, this is a standard integral, and the result is: $$E[X] = \frac{1}{2}$$ ### Step 3: Compute $$\text{Var}(X)$$ Variance is defined as: \[ \text{Var}(X) = E[X^2] - (E[X])^2 \] Substituting the values: $$\text{Var}(X) = \frac{3}{2} - \left( \frac{1}{2} \right)^2 = \frac{3}{2} - \frac{1}{4} = \frac{5}{4}$$ ### Final Answers: - $$E[X^2] = \frac{3}{2}$$ - $$\text{Var}(X) = \frac{5}{4}$$ Would you like further explanation on any step or additional details? Here are some related questions for practice: 1. How would you find $$E[X^3]$$ for this distribution? 2. What is the cumulative distribution function (CDF) of this random variable? 3. How would you verify that $$f(x)$$ is a valid probability density function? 4. Can you calculate the skewness of this distribution? 5. How does the expected value change if the pdf is modified to $$f(x) = \lambda e^{-\lambda x}$$? **Tip**: Always check that the total integral of a pdf equals 1 to ensure it's properly normalized.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability Theory
Continuous Random Variables
Expectation
Variance
Integration
Formulas
E[X^2] = ∫ x^2 f(x) dx
Var(X) = E[X^2] - (E[X])^2
f(x) = 2e^(-2x), x > 0
Theorems
Properties of Exponential Distributions
Integration by Parts
Gamma Function
Suitable Grade Level
Undergraduate
Related Recommendation
Find E[X^2] and Var(X) for Exponential Distribution with PDF f(x) = 2e^(-2x)
Calculate Mean and Variance of a Continuous Random Variable with Exponential PDF
Finding Mean and Variance for an Exponential Distribution
Calculating Expected Value and Variance for Exponential Distribution f(x) = 4e^(-4x)
Find Expected Value and Variance of a Random Variable with Piecewise PDF