The expected value of a discrete variable is defined as:

E(X) = ∑x*P(X=x) (for all possible values of x, in this case x = {1,2,3,4,5} )

(next, we will expand the summation and substitute in the values given, we are able to find P(X=x) simply by substituting in the equation from above)

= 1*(0.09+0.011^2) + 2(0.09+0.012^2) + 3(0.09+0.013^2) + 4(0.09+0.014^2) + 5(0.09+0.01*5^2)

(next, we will factorize in order to make calculation easier and less prone to mistakes)

= 0.09(1+2+3+4+5) + 0.01*(11^2 + 22^2 + 33^2 + 44^2 + 5*5^2)

(next, we will add up the first bracket and simplify the second bracket)

= 0.09*(15) + 0.01*(1^3 + 2^3 + 3^3 + 4^3 +5^3)

(next, we will multiply through the first bracket and multiply the individual expressions in the second bracket)

= 1.35 + 0.01*(1 + 8 + 27 + 64 + 125)

(next, we add up the second bracket)

= 1.35 + 0.01*(225)

= 1.35 + 2.25 = 3.6

E(X) = 3.6 (the expected value of X is 3.6)

differentiate 4x^3 + 3x