Word Count : 387

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The given probability distribution is: \[ P(X = x) = \frac{2}{3} \left( \frac{1}{3} \right)^x, \quad x = 0, 1, 2, \dots \] We need to find the moment generating function (MGF), mean, and variance of \(X\). ### Moment Generating Function (MGF): The MGF of a random variable \(X\) is defined as: \[ M_X(t) = E[e^{tX}] = \sum_{x=0}^{\infty} e^{tx} P(X = x) \] Substitute the given distribution into the equation: \[ M_X(t) = \sum_{x=0}^{\infty} e^{tx} \left( \frac{2}{3} \left( _________ ____ ___ _______ _____ ______ ______ _________ ________ ______.
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