Solution explain:

Example 35: We consider first a standard fair die with 6 sides. For such a die the 6 expectation is given by E(X) = _ k. = = 3.5 and the variance is V( X) = k=1 6 C (k - 3.5) .- = 2.9. Now let us consider the second fair die where k=1 number on sides 3 and 4 are substituted by 1 and 6 (so the die has sides 1, 2, 1, 6, 5, 6). For this die the expectation is 3.5 as before but the variance is much larger, V(Y) = 4.9. Distribution of both dice are shown on the figure below (blue colors) together with the corresponding standard deviations (green colors give expectation +/ - standard deviation)