Math Problem Statement
Let X be a continuous random variable with probability mass function{휆푒−휆푥,푥>00,표푡ℎ푒푟푤푖푠푒.i.Find its moment generating function if it exists
Solution
To determine the moment generating function (MGF) of the given continuous random variable , we start with the provided probability density function (PDF):
\begin{cases} \lambda e^{-\lambda x}, & x > 0, \\ 0, & \text{otherwise}. \end{cases}$$ The moment generating function $$M_X(t)$$ is defined as: $$M_X(t) = \mathbb{E}[e^{tX}] = \int_{-\infty}^\infty e^{tx} f_X(x) \, dx.$$ ### Step 1: Substitute the PDF into the integral Since $$f_X(x) \neq 0$$ only for $$x > 0$$, the limits of integration reduce to $$0$$ to $$\infty$$: $$M_X(t) = \int_0^\infty e^{tx} \lambda e^{-\lambda x} \, dx.$$ Simplify the exponent by combining terms: $$M_X(t) = \lambda \int_0^\infty e^{-(\lambda - t)x} \, dx.$$ ### Step 2: Conditions for convergence The integral converges only if $$\lambda - t > 0$$, i.e., $$t < \lambda$$. If this condition is not met, $$M_X(t)$$ does not exist because the integrand grows unbounded. ### Step 3: Evaluate the integral Assuming $$t < \lambda$$, the integrand becomes $$e^{-(\lambda - t)x}$$. The integral of an exponential function is: $$\int_0^\infty e^{-(\lambda - t)x} \, dx = \frac{1}{\lambda - t}.$$ Thus: $$M_X(t) = \lambda \cdot \frac{1}{\lambda - t}.$$ ### Step 4: Final expression The moment generating function of $$X$$ is: $$M_X(t) = \begin{cases} \frac{\lambda}{\lambda - t}, & t < \lambda, \\ \text{undefined}, & t \geq \lambda. \end{cases}$$ --- ### Key Insights: - The MGF $$M_X(t)$$ exists only for $$t < \lambda$$. - The domain restriction ensures that the integral converges. Would you like to explore properties derived from this MGF (e.g., moments) or verify related results? --- ### 5 Related Questions: 1. What is the first moment (mean) of $$X$$ based on its MGF? 2. Can we derive the variance of $$X$$ from the MGF? 3. How does $$\lambda$$ influence the shape and spread of the distribution? 4. What is the significance of the convergence condition $$t < \lambda$$? 5. Can you derive the cumulative distribution function (CDF) of $$X$$? ### Tip: The MGF is a powerful tool for finding moments like the mean and variance of a random variable, especially for exponential and normal distributions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Moment Generating Functions (MGFs)
Exponential Distribution
Formulas
Moment Generating Function: M_X(t) = E[e^(tX)] = ∫ e^(tx) f_X(x) dx
Exponential Integral: ∫ e^(-ax) dx = 1/a, for a > 0
Theorems
Definition of Moment Generating Function
Convergence Criterion for Exponential Integrals
Suitable Grade Level
Undergraduate (Probability and Statistics)
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