Find the expected value E(X), the variance Var(X) and the...

Find the expected value E(X), the variance Var(X) and the standard deviation ?(X) for the density function. (Round your answers to four decimal places.) f(x) = e^x on [0, ln 2] E(X) = .3863 Var(X) = ?(X) =

Transcript

00:01 Hello welcome to this lesson in this lesson we have the density function f of x equals to e to the power x and this is on zero length two so we are looking at the density function right and this would find the expected value or the expectation of x will find the variance and the standard deviation so the expectation of x is equals to the integral of the bounce from a to b then x times f of x dx right so here we are looking at from zero to line of two then x e to the power x dx so this would evaluate using integration by path by parts so here would have using the integration by parts let me actually not use the bounds now so if we have x e to the power x dx we can write this thing as integral u dv and this using the integration by parts would have u times v minus the integral v times dv so at this point we let u equals to x which implies that du is equals to dx so this u is equals to x and now we have dv okay this dv also equals to e to the power x dx which also implies that v is equals to e to the power x and at this point we would have equals to u being so we have x e to the power x then minus the integral v okay du which is cx so this becomes e x times e to the power x minus e to the power x okay so at this point let's evaluate it at the bounds and this would have from zero to lane right from zero to lane two so this would actually become if i put in link to have lane two e to the power lane two minus e to the power lane two then if i put in zero i have zero e to the power zero minus e to the power zero and this is the whole of this reduces to two so two lane two then minus the whole of this results okay then minus this goes to zero and this goes to one so half plus one and now the expectation expected value will become two lane two minus one so two before the small basis we would have 0 .8 0 .38635 and now let's look at the variance of x which is equals to the integral from a to b x squared f of x dx minus expectation of x squared so this petition is what found as this all right so now we'll have from zero to lane two then x squared e to the power x dx then minus the expectation squared all right so at this point we are dealing with only x squared e to the power x dx using the…

Question

Find the expected value E(X), the variance Var(X) and the standard deviation ?(X) for the density function. (Round your answers to four decimal places.) f(x) = e^x on [0, ln 2] E(X) = .3863 Var(X) = ?(X) =

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