Question

Let \(X_{1}, X_{2}\) be two independent random variables having gamma distributions with parameters \(\alpha_{1}=3, \beta_{1}=3\) and \(\alpha_{2}=5, \beta_{2}=1\), respectively. (a) Find the mgf of \(Y=2 X_{1}+6 X_{2}\). (b) What is the distribution of \(Y ?\)

Short answer

The MGF of \(Y = 2X_1 + 6X_2\) is given by \((1 - t/6)^{-3} * (1 - 3t/2)^{-5}\), for \(t < 2/3\), which implies that the random variable \(Y\) follows a gamma distribution.

Step-by-step solution

- Compute the MGF of the initial variables

To find the moment generating function (mgf) of \(Y\), we need to first know the mgfs of \(X_1\) and \(X_2\). The mgf of a gamma distribution with parameters \(\alpha\), \(\beta\) is given by \((1 - t/\beta)^{-\alpha}\) for \(t < \beta\). Thus, the mgfs of \(X_1\) and \(X_2\) are \((1 - t/3)^{-3}\) and \((1 - t)^{-5}\), respectively.

- Apply MGF properties for linear transformations and sum of independent variables

The mgf of a linear combination of independent random variables is the product of the mgfs of the transformed variables. Here, \(Y = 2X_1 + 6X_2 = 2(X_1 + 3X_2)\). The coefficient 2 has effect of dividing the argument of the mgf by 2, while the sum is reflected by multiply the mgfs. So, the mgf of \(Y\) will be as follows: \[M_Y(t) = M_{X_1}(t/2) * M_{X_2}(3t/2) = (1 - t/6)^{-3} * (1 - 3t/2)^{-5}\] for \(t < 2/3\).

- Identify the distribution

The distribution of random variable \(Y\) is identified by matching the mgf obtained to that of known distributions. Due to independence and identical distribution properties, \(Y\) holds gamma distribution with combined parameters.