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

\[
f(x) = 
\begin{cases}
\frac{1}{2 \sqrt{x}} & 0 < x < 1 \\
0 & \text{otherwise}
\end{cases}
\]

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.

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

This page explores how to find the moment-generating function (MGF) of a random variable X whose probability density function (PDF) is given by f(x) = 1/(2√x), 0 < x < 1. It covers key concepts in probability theory and integration.

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