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I know that when if we have an exponential random variable with parameter $\lambda$, the moment generating function is $\frac{\lambda}{\lambda-t}$ when $t < \lambda$, but what can I say about the function when $t \ge \lambda$? Based on my computation of the integral, I think it is $+\infty$, but I'm not too sure about this. Thank you

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It is $\infty$ becasue $\lambda \int_0^{\infty} e^{(t-\lambda)x} dx \geq \lambda \int_0^{\infty} 1 dx=\infty$.

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