Probability Density Functions, Expectations and Moments

Transcription

1 Probability Density Functions, Epectations and Moments PROBABILITY Measurement systems have random variates as their numerical output values. The set of all possible outputs is the sample space for the measurement system and is denoted by a lower-case. In general, the set of outputs ranges between min and ma and, in many common cases, it happens that min = - and ma =. Thus the range of possible eperimental numerical results is min ma. Suppose that i is the numerical result of the i-th measurement, of N total measurements, and that a and b are possible numerical results, with min a < b ma.. Let N ab denote the number of times that measured results (i.e., y i values) are greater than or equal to a, and less than or equal to b. In other words, N ab is the number of times, out of N total times, that measured results fall within the closed interval [a,b]. Then f(a i b) is the eperimental relative frequency with which measured results, from N eperiments, fall within the closed interval [a,b] and f(a i b) is defined by f( a b) N / N i ab The probability that a measured result i falls within the closed interval [a,b] is denoted by P(a i b) and is defined as Pa ( i b) lim N ( Nab / N) 1

2 Thus, P(a i b) is the limiting value, for an infinite number of measurements, of f(a i b). Clearly, 0 P(a i b) 1. As a special case of the notation for P(a i b), note that P( i ) is simply the probability that a measured result i is less than or equal to a specified value. THE CUMULATIVE DISTRIBUTION FUNCTION The cumulative distribution function (CDF) for the random variate is CDF F () P( i ) Thus, F () is the probability that a measured result i is less than or equal to the quantity. Clearly, F ( min ) = 0 F () 1 = F ( ma ) If min = - and ma =, then F (- ) = 0 F () 1 = F ( ). If 1 < 2, then F ( 1 ) F ( 2 ). Thus, F () never decreases as increases. THE PROBABILITY DENSITY FUNCTION The probability density function (PDF) is defined as the derivative of the CDF: PDF p () df ()/d, so PDF = d(cdf)/d. 2

3 Then, F ( ) = 0 implies F ( ) = p ( λ) dλ = P( ) min i min Therefore, the area under p (), from min to, is the probability that an eperimental result i is less than or equal to the value. PROPERTIES OF THE PROBABILITY DENSITY FUNCTION 1. p () = df ()/d 0 non-negative probability ma 2. p( ) d = 1 normalized to unity area min 3. P( a X b) = F( b) F( a) = p( ) d a b If a and b + d, then p ()d = P( X + d), so p () is a probability density. As a consequence, the PDF is also called the probability density function. Indeed, this is the customary reading of PDF. Similarly, the CDF is the cumulative density function. MOMENTS AND MEANS The statistical (population) mean value µ, of a random variable, is the numerical average of the values that can assume, weighted by their probabilities. Mean values are also called epected or epectation values. Thus, for a continuous PDF given by p (), the epectation value of, denoted E[], is defined as 3

4 E[ ] µ p ( ) d if the integral eists, i.e., does not diverge. The integration limits are more generally given as min and ma, but no harm is done in using ± integration limits because integration over regions where p () is identically zero contribute nothing to the epectation value of. For a discrete PDF, p () is simply of the following form: j P ( ) δ( ) j j If is a random variate and g( ) is any analytic function, then g() is a random variate with mean value E[g()], given by Eg [ ( )] µ g ( ) p( d ) g( ) If g() n n, then µ g() is the n-th moment of. If g() ( µ ), then µ g() is the n-th central moment of. SPECIAL MOMENTS 1. If n = 1, then g() =, so µ g() = µ. Thus, the first moment of is the population mean of. 2. If n = 2, then g() = 2. Thus, the second moment of is the population mean-square of. 4

5 3. The population variance of is the second central moment of. Thus, population variance σ E[( µ ) ] ( µ ) p ( ) d The population standard deviation of is σ, i.e., the positive square root of the population variance. An important consequence: Since ( µ ) = 2 µ + µ, we have σ 2 = E [ 2 2µ + µ 2 ] = µ 2µ 2 + µ 2 = µ µ σ = µ µ 2 Thus, the variance is the mean-square minus the squared mean. Measurement system interpretation: Suppose that is a time-variant voltage, i.e., v(t), eactly as is case for the output of a typical modern electronic measurement system. Then: 1. µ = µ = mean voltage = DC component of v(t) units: Volts this is the constant (i.e., time-independent) component of v(t) µ = µ = squared mean voltage = DC power of v(t) units: Volts squared this is the power in the DC component of v(t) 3. µ 2 = µ 2 = mean square voltage = total average power of v(t) units: Volts squared v () t 5

6 σ = σ = variance = AC power of v(t) = µ 2 µ units: Volts squared v () t vt () AC power of v(t) = total average power of v(t) - DC power of v(t) 5. σ σ = = standard deviation = + variance units: Volts this is the root mean square of the time-varying component of v(t), i.e., the rms value of the AC component of v(t). If µ = µ = 0, either intrinsically or as a result of DC subtraction, then 2 σ = σ v t = + µ 2 = + µ 2 = positive square root of the mean of v t v t () () () 6. Any time-varying waveform can be resolved into two components: a constant plus a zero-mean, time-varying component. The constant is simply the mean value of the original time-varying waveform. The variance of the time-varying waveform resides completely in the zero-mean, time-varying component of the original time-varying waveform. Thus, the mean and variance are separable. 7. A set of measurement values, i.e., results from replicate measurements, may be considered as sequential samples of a fictitious time-varying waveform. Hence, the set of values will have a mean value and a fluctuating component that entirely accounts for the observed variance of the set of values. 8. Note that power, as used above, has units of volts squared, i.e., V 2. Physically, of course, power has units of watts. However, we are ignoring all impedances, which have units of ohms, because it is only the magnitudes of the measured voltage values that are relevant. This is the conventional fictitious one ohm resistance assumption. 6