2.1 Properties of PDFs

Transcription

1 2.1 Properties of PDFs mode median epectation values moments mean variance skewness kurtosis 2.1: 1/13

2 Mode The mode is the most probable outcome. It is often given the symbol, µ ma. For a continuous random variable, the mode is most often obtained by taking the derivative of the pdf and setting it equal to zero. Neither eample we have been using (rolling two dice and fluorescence decay) are amenable to the derivative approach. The two modes are shown below. mode τ = 5 ns f().1.5 f(t) : 2/ t (ns)

3 Median The median is the value of the random variable yielding F(m) =.5. It is often given the symbol µ 1/2. For a discrete cdf there may not be a value of the random variable that eactly produces.5. Rolling two dice: ( ) 12 ( m+ 6)( m+ 7) 2 F m = =.5 m + 13m+ 6 = 72 µ = = = Fluorescence decay: ( ) ( m ) = ( ) = F m µ 12 = 1 ep 5 =.5 5 ln f(t) median τ = 5 ns 2.1 : 3/ t (ns)

4 Epectation Values Consider a random variable,, and some arbitrary function of the random variable, y(). The epectation value of the function is a weighted average of y() taken over all values of the random variable. The weighting factor is the probability of observing a given value of. 2.1 : 4/13 For a discrete random variable, the epectation value of y() is given by a summation. ( ) ( ) ( ) y f ( ) ( ) ( ) ( ) ( ) E y = = y f = y p f For a continuous random variable, the epectation value of y() is given by an integral. ( ) ( ) ( ) y f ( ) d ( ) ( ) E y = = y f d f d

5 Mean The mean is the epectation value of the random variable, i.e. y() =. The mean has units of. For a discrete random variable, the mean is given by a summation. µ E( ) = f ( ) = p( ) For a continuous random variable, the mean is given by an integral. ( ) ( ) µ = E f d When a pdf is due to noise, the mean is associated with the true value. That is, as the noise approaches zero the measured data will cluster closer to the mean. When a pdf is due to a signal, the mean may not signify anything. The mean of a fluorescence spectrum has no significance, but the mean of a chromatographic peak is the retention time. 2.1 : 5/13

6 Mean for Rolling Two Dice The pdf for rolling two dice and subtracting the "blue" value from the "red" value is given by a two-part function. ( ) ( ) ( ) ( ) f = f = 6 36 < 5 When writing the mean, the two part function is accommodated by using summations with different sets of limits. ( 6+ ) ( 6 ) 5 µ = To simplify, use the fact that Σ[φ 1 ()+φ 2 ()] = Σφ 1 () + Σφ 2 () µ = + + = : 6/13

7 Mean for Fluorescence Decay The pdf for a 5-ns fluorescence decay is given by the following. 1 ep 5 5 ( ) = ( t ) f t The mean is given by the epectation value of t. 1 = ep( 5) 5 t t dt µ Solve for the mean using the following tabulated definite integral, where a = 1/5. 2 ( ) = ep a d 1 a 2 15 µ = = 5 51 note that the mean is equal to the decay constant τ (lifetime) 2.1 : 7/13

8 Moments The deviations of a random variable are given by, d i = i - µ. Note that the epectation value of the deviations is zero. ( ) ( ) ( ) ( ) ( ) E d = µ p = p µ p = µ µ = i i i i i i i i ( ) ( ) ( ) ( ) or E d = E µ = E E µ = µ µ = An epectation value of the form, E[ n ], is called the n th moment. The mean is the first moment. An epectation value of the form, E[d n ] = E[(-µ) n ], is called the n th moment about the mean. As will be seen, the variance is the second moment about the mean. The mean, mode and median are used to locate the probability density function along the random variable ais, while moments are used to characterize its shape. 2.1 : 8/13

9 Variance Variance is the second moment about the mean i µ i i i µ i ( ) ( ) ( ) ( ) V = p = p i ( ) ( ) ( ) ( ) V = µ f d = f d µ The standard deviation, σ, is the square root of the variance, σ 2 = V(). The standard deviation is a measure of the spread of probability along the random variable. The standard deviation has units of. When a pdf is due to noise, the standard deviation is used as a quantitative measure of the noise. When the pdf is due to a signal, the standard deviation is often a measure of resolution. The relative standard deviation is a measure of spread normalized by the mean. This avoids problems caused by scaling. RSD = σ/µ RSD is unitless 2.1 : 9/13

10 Variance for Rolling Two Dice When writing the variance, the two-part function is accommodated by using summations with different sets of limits. σ σ σ ( 6+ ) ( 6 ) = = = Use, n ( + 1)( n ) ( + 1) 2 nn n nn = = to obtain: σ 2 ( 5 6) = = σ = : 1/13

11 Variance for Fluorescence Decay The variance is given by the second moment about the mean. σ ( ) = ep t t dt Solve for the variance using the following tabulated definite integral, where a = 1/5. σ ( ) 2 3 ep a d = 2 a σ = = = : 11/13

12 Skewness Skewness is a measure of pdf asymmetry. It can be negative (skewed to the left), positive (skewed to the right), or zero (symmetric about the mean). Skewness is given by the third moment about the mean. 3 3 i 3 3 i i i i i ( ) f ( ) f ( ) µ = µ = σ µ µ ( ) ( ) ( ) µ = µ f d = f d 3σ µ µ Tabulations often list the coefficient of skewness, α = µ σ 3 3 Skewness for the dice is zero since the pdf is symmetric. For the fluorescence decay, the coefficient of skewness is given by, 3 α t ep ( t 5 ) dt = = : 12/13

13 Kurtosis Kurtosis is a measure of how fast probability drops as the random variable moves away from the mean. The larger the value, the more rapid the drop. Its primary use is in identifying a normal pdf which has a coefficient of kurtosis of 3. Kurtosis is given by the fourth moment about the mean = ( i ) f ( i) = i f ( i) i i = ( ) f ( ) d = f ( ) d µ µ µ µ σ µ µ µ µ µ µ σ µ µ Tabulations often list the coefficient of kurtosis. α = µ σ 4 4 The coefficient of kurtosis for rolling two dice is For the fluorescence decay it is : 13/13