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okay so were going to have a look at this nice formula for the expectation of a continuous random variable where if youve got a non-negative random variable you can write the expectation as this integral thats greater than or equal to s with respect to s so if youve seen the case for a discrete random variable the proof is going to be quite similar here basically well write this the expectation as an integral using the definition and well write this as two integrals change the order of integration then out comes the formula so to get started well have a look at this in the case where youve got a pdf and well have a look at the more general version using measure theory at the end briefly so you need a non-negative continuous random variable and say its pdf is this function f x of x so just by definition the expectation of our random variable x the integral of x multiplied by the pdf with respect to x over this range of values that x can take and then the trick here is just to