The Sample Space

The Sample Space 

Definition. With each experiment ε of the type we are considering we define the sample space as the set of all possible outcomes of ε. We usually designate this set by S. (In our present context, S represents the universal set described previously.) Let us consider each of the experiments above and describe a sample space for each. The sample space S_i will refer to the experiment E_i
S₁: { l,2,3,4,5,6}.
S₂: {0, 1, 2, 3, 4}. The Sample Space 9
S₃: {all possible sequences of the form a₁, a₂, a₃, a₄, where each a₁ = H or T depending on whether heads or tails appeared on the i^th toss.}
S₄: {0, 1, 2, ... , N}, where N is the maximum number that could be produced in 24 hours.
S₅: {0, I, 2, ... , M}, where M is the number of rivets installed.
S₆: {t | t ≤ 0}.
S₇: {3, 4, 5, 6, 7, 8, 9, 10}.
S₈: {10, 11, 12, ... }.
S₉: {v_x , v_y, v_z | v_x, v_y , v_z real numbers}.
S₁₀: {h i . ... ' hn I hi ≥ 0, i = 1, 2, ... 'n}.
S₁₁: {T | T ≥ O}.
S₁₂: {black ball}.
S₁₃: This sample space is the most involved of those considered here. We may realistically suppose that the temperature at a specified locality can never get above or below certain values, say M and m. Beyond this restriction, we must allow the possibility of any graph to appear with certain qualifications. Presumably the graph will have no jumps (that is, it will represent a continuous function ). In addition, the graph will have certain characteristics of smoothness which can be summarized mathematically by saying that the graph represents a differentiable function. Thus we can finally state that the sample space is
 {f | fa differentiable function, satisfying m ≤ f(t) ≤ M, all t}.

S₁₄: { (x, y) | m ≤  x ≤  y ≤  M}. That is, S₁₄ consists of all points in and on a triangle in the two-dimensional x,y-plane.
(In this book we will not concern ourselves with sample spaces of the complexity encountered in S₁₃. However, such sample spaces do arise, but require more advanced mathematics for their study than we are presupposing.)
In order to describe a sample space associated with an experiment, we must have a very clear idea of what we are measuring or observing. Hence we should speak of "a" sample space associated with an experiment rather than "the" sample space. In this connection note the difference between S₂ and S₃• Note also that the outcome of an experiment need not be a number. For example, in E₃ each outcome is a sequence of H's and T's. In E₉ and E₁₀ each outcome consists of a vector, while in E₁₃ each outcome consists of a function.
It will again be important to discuss the number of outcomes in a sample space. Three possibilities arise: the sample space may be finite, countably infinite, or noncountably infinite. Referring to the above examples, we note that S₁, S₂, S₃, S₄, S₅, S₇, and S₁₂ are finite, S₈ is countably infinite, and S₆, S9, S₁₀, S₁₁, S₁₃, andS₁₄ are noncountably infinite. At this point it might be worth while to comment on the difference between a mathematically "idealized" sample space and an experimentally realizable one. For this purpose, let us consider experiment E₆ and its associated sample space S₆  . It is clear that when we are actually recording the total time t during which a bulb is functioning, we are "victims" of the accuracy of our measuring instrument. Suppose that we have an instrument which is capable of recording time to two decimal places, for example, 16.43 hours. With this restriction imposed, our sample space becomes countably infinite: {0.0, 0.0 1, 0.02, . .. } . Furthermore, it is quite realistic to suppose that no bulb could possibly last more than H hours, where H might be a very large number. Thus it appears that if we are completely realistic about the description of this sample space, we are actually dealing with a finite sample space: {0.0, 0.0 1, 0.02, .. . , H}. The total number of outcomes would be (H / 0.0 1) + 1, which would be a very large number if His even moderately large, for example, H = 100. It turns out to be far simpler and convenient, mathematically, to assume that all values of t  ≥  0 are possible outcomes and hence to deal with the sample space S₆ as originally defined. In view of the above comments, a number of the sample spaces described are idealized. In all subsequent situations, the sample space considered will be that one which is mathematically most convenient. In most problems, little question arises as to the proper choice of sample space.