 Hi everyone, it's MJ and we're looking at a sample question from exam P and it says the value of a piece of factory equipment after three years of use is 100 times 0.5 to the power of X Where X is a random variable having the moment generating function Given over here, and it says calculate the expected value of this piece of equipment after three years of use Now this question at first seems very very Intimidating like what moment generating functions. This is intense, but let's go back to the theory So we would have learned in this course that the moment generating function is Given by the following. It's the expected value of E Tx, okay. I think we're all happy with that So what we can do what we can do is if we look at the Expected value so the expected value of this piece of equipment Okay, which is given by this over here if we do this 0.5x We see straight away. We can just take that 100 out. So we have the expected of 0.5x Now this is where the big trick comes in. How do we write this in? This format so that we can use our moment generating function And this is where you do need to be a little bit comfortable with how exponentials and Logarithms work. So what we're going to be doing here in this next step is we're going to say a hundred e Expected and we're gonna have you e x Lin 0.5 Okay, and that is a bit of a mathematical jump to go from there to there But we haven't changed anything. We're just representing it in a different way Because remember the exponential of Lin 0.5 is the same as saying 0.5 to the power of x Okay, but what this does do by writing it in this in this way We can see that hold on hold on we now have our moment generating function format, which means we are equal to a hundred Mx Lin 0.5 and We just now use this formula that they've given us over here To get the following we have a hundred times one Divide about one minus two and in this case our t is Lin 0.5 and If we put that into the calculator, we can get forty one point nine. We look is forty one point nine one of our options Yes, it is and we have the answer So the big trick in this question was getting us from this step to this step over here so that we could then rely on the moment generating function and you would have known to do that if you Were comfortable and you had learned what your moment generating functions represent Anyway, if you've got any questions, please feel free to let me know in the comment section below. Cheers