Math Problem Statement

Find the MGF of a random variable X whose density function is f(x) = 1/(2√x), 0 < x < 1, 0 otherwise.

Solution

To find the moment-generating function (MGF) of a random variable XX whose probability density function (PDF) is given as:

\begin{cases} \frac{1}{2 \sqrt{x}} & 0 < x < 1 \\ 0 & \text{otherwise} \end{cases}$$ The moment-generating function (MGF), $$M_X(t)$$, is defined as: $$M_X(t) = \mathbb{E}\left[e^{tX}\right] = \int_{-\infty}^{\infty} e^{tx} f(x) \, dx$$ For this particular PDF, we only need to integrate from 0 to 1, since $$f(x) = 0$$ outside this interval. Thus, the MGF is: $$M_X(t) = \int_0^1 e^{tx} \cdot \frac{1}{2 \sqrt{x}} \, dx$$ Let's now proceed to solve this integral to find the MGF.The moment-generating function (MGF) for the given random variable $$X$$ is: $$M_X(t) = \frac{-i \sqrt{\pi} \, \text{erf}(i \sqrt{t})}{2 \sqrt{t}}$$ Here, $$\text{erf}(z)$$ is the error function, and $$i$$ represents the imaginary unit. This expression can be simplified further for specific values of $$t$$, but this is the general form of the MGF. Would you like to explore more details about the steps or evaluate this function for specific values of $$t$$? Here are 5 related questions: 1. What is the physical interpretation of the moment-generating function (MGF)? 2. How can we use the MGF to find moments such as the mean or variance of $$X$$? 3. What is the role of the error function ($$\text{erf}$$) in probability theory? 4. Can we approximate or simplify the MGF for small values of $$t$$? 5. How can we verify if this distribution is valid (i.e., does the PDF integrate to 1)? **Tip:** The MGF is useful for deriving all the moments of a distribution (mean, variance, etc.), as it encodes them within its series expansion.

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Moment-Generating Functions (MGF)
Integration

Formulas

MGF M_X(t) = E[e^{tX}] = ∫_{-∞}^{∞} e^{tx} f(x) dx
f(x) = 1/(2√x), for 0 < x < 1

Theorems

Moment-Generating Function Theorem
Fundamental Theorem of Calculus

Suitable Grade Level

University (Undergraduate)