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Similarly, a current source requires that a direction be indicated, as well as the function, as shown in Fig. 2-4(b). For passive circuit elements R, L, and C, shown in Fig. 2-4(c), the terminal where the current enters is generally treated as positive with respect to the terminal where the current leaves. The sign on power is illustrated by the dc circuit of Fig. 2-5(a) with constant voltage sources VA 20:0 V and VB 5:0 V and a single 5- resistor. The resulting current of 3.0 A is in the clockwise direction. Considering now Fig. 2-5(b), power is absorbed by an element when the current enters the element at the positive terminal. Power, computed by VI or I 2 R, is therefore absorbed by both the resistor and the VB source, 45.0 W and 15 W respectively. Since the current enters VA at the negative terminal, this element is the power source for the circuit. P VI 60:0 W con rms that the power absorbed by the resistor and the source VB is provided by the source VA .

It follows from the above that if X is a continuous random variable, then the probability that X takes on any one particular value is zero, whereas the interval probability that X lies between two different values, say, a and b, is given by

EXAMPLE 2.4 If an individual is selected at random from a large group of adult males, the probability that his height X is precisely 68 inches (i.e., 68.000 . . . inches) would be zero. However, there is a probability greater than zero than X is between 67.000 . . . inches and 68.500 . . . inches, for example.

CHAP. 2]

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A function f (x) that satisfies the above requirements is called a probability function or probability distribution for a continuous random variable, but it is more often called a probability density function or simply density function. Any function f (x) satisfying Properties 1 and 2 above will automatically be a density function, and required probabilities can then be obtained from (8).

Not all statistics are appropriate for all scales. The following indicates which common statistical methods are appropriate: Nominal scale: Only mode, median, and percentiles Ordinal scale: The above and Spearman correlations Interval scale: The above and mean, standard deviation, and Pearson correlations Ratio scale: All statistics Absolute scale: All statistics

Fig. 2-5

f (x)

is a density function, and (b) compute P(1 (a) Since f (x) satisfies Property 1 if c X 2).

The passive circuit elements resistance R, inductance L, and capacitance C are de ned by the manner in which the voltage and current are related for the individual element. For example, if the voltage v and current i for a single element are related by a constant, then the element is a resistance, R is the constant of proportionality, and v Ri. Similarly, if the voltage is the time derivative of the current, then the element is an inductance, L is the constant of proportionality, and v L di=dt. Finally, if the current in the element is the time derivative of the voltage, then the element is a capacitance, C is the constant of proportionality, and i C dv=dt. Table 2-1 summarizes these relationships for the three passive circuit elements. Note the current directions and the corresponding polarity of the voltages.

1 > 9. 2)

ohms ( )

x3 2 27 1

8 27

v Ri (Ohms s law)

Temperature is an interval scale measure. Thus, it makes statistical sense to give an average temperature. However, the numbers on baseball players uniforms are a nominal scale measure. It does not make sense to give the average of the numbers on a team s uniforms. Similarly, the average ranking of the students in a class or the average of a student s rankings in a number of classes is not appropriate.

1 27

henries (H)

7 27

In case f (x) is continuous, which we shall assume unless otherwise stated, the probability that X is equal to any particular value is zero. In such case we can replace either or both of the signs in (8) by . Thus, in Example 2.5, 7 P(1 X 2) P(1 X 2) P(1 X 2) P(1 X 2) 27

farads (F)

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