A probability experiment consists of measuring a voltage val

A probability experiment consists of measuring a voltage value, say V, in the range -10 to +10 volts, inclusive. What is the sample space, S, for the experiment? (b) Suppose that the random voltage in part (a) is input to a half-wave rectifier before we take the measurement. The rectifier simply takes the absolute value of the variable V: If the probability experiment consists of measuring a value of X (rather than V), what is the sample space, say S_x? (c) Which sample space above, S, or S_x, contains more points? (d) Repeat parts (a) and (b) if -3 volts is added to the variable V before any measurements are taken.

Solution

Note: Throughout we distinguish between the continuous (C) and the discrete (D) cases.(it is not specified in the problem)

In the discrete case, it is assumed that the voltages are integral points .

I. The continuous case (C). In this case the voltage assumes all real numbers in the range specified.

(a) The sample space S is the set of all points in the interval [-10,10] = { x , x real number lying between -10 and 10}

(b) As only absolute values are involved, the sample space S_X is the interval [0,10]= {x , x , positive, real number lying between 0 and 10}

(c) In the continuous case, both sample spaces contain infinite number of points

(d) S consists of all points in the closed interval [-13,7]

   S_X consists of all points in the closed interval [0,13]

2. Discrete case (D): It is assumed that voltages assume integer values

(a) S = { -10,-9,-8,.......0,1,2.....9,10}

(b) S_X = {0,1,2,3,....9,10}

(c) S contains more points.

(d) S = {-13,-12,-11,.....0,1,2,...7}

S_X = {0,1,2,....13}