Module 3 Statistics I. Basic Statistics A. Statistics and Physics 1. Why Statistics Up to this point, your courses in physics and engineering have considered systems from a macroscopic point of view. For instance, we have described baseballs, blocks, airplanes, etc. as rigid bodies. In our discussion of the kinetic theory of gases in University Physics, we demonstrated that the macroscopic properties of a gas including temperature, pressure, and volume are derived from the microscopic motion of the molecules that compose the gas. Since all matter is composed of atoms, we should epect that this approach is universal and would provide a greater understanding than just studying macroscopic properties (the average of microscopic properties)! This particular field of physics is called Statistical Mechanics since it combines mechanics (classical or quantum) and statistics. In statistical mechanics and quantum mechanics, we talk about calculating the probability that a particle has some physical attribute. The attribute might be energy, linear momentum, position, etc. In the discussion below, we will consider the attribute to be position, but any attribute could be inserted. The reason for choosing position is to simplify the tet and because position is easier to visualize. Humans are equipped with position detectors called eyes.. Probability Distribution Function - f () The probability distribution function is a function that determines the probability that an object is located between and + d. It is defined as the change in probability function over the change in and is most useful in dealing with problems continuous physical quantities. () ΔP() Δ f "Discrete Variable " () dp() d f "Continuous Variable "
3. Calculating Probabilities Using the Probability Distribution Function Given the probability distribution function, we can calculate the probability that a particle is located in the region between i and f by f P(1 ) f ( )Δ ΔP( ) "Discrete Variable " k i k P(1 ) f i f k i k f ()d "Continuous Variable " 4. Normalization If we consider all possibilities, we will have a 100% probability of finding the particle. Therefore, the sum/integral of the probability distribution function over all possible values of must equal one!! P ( ) P( ) 1 "Discrete Variable " P( ) k k f () d "Continuous Variable " 1 If the function f() doesn't have this property then it is said to be unnormalized and can NOT be a probability distribution function! In order to create a probability distribution function, we divided the function by the result of the previous equation. This process is called normalization! 5. Calculating the Average (Epectation) Value - or The average location of the object can be calculated using ΔP() "Discrete" f ()d "Continuous"
In physics, the average value of a physical quantity is usually called its "epectation value." This is because it is the value that is epected on average from multiple measurements of the quantity even though no single measurement may give this value. Consider a class in which half the students score 100 and the other half score 0 on a test. The class average (epectation value) is 50% even though no individual student had this result! Mathematicians call this type of average value the "mean." 6. Standard Deviation () and Variance ( ) Although the epectation value is important, it doesn't completely specify how a system is behaving. For eample, the average voltage out of a wall plug is zero (no DC voltage). However, the standard deviation is 110 volts!! Obviously, the standard deviation is important since this is what makes your TV, radio, and other appliances work! The variance and standard deviation tells us how much the location of the particle will vary (i.e. the spread) on average as we make several measurements. Consider the following graph. Both the red system and blue system have the same average. Average Measurement From the definition of the average, we know that if we sum the distance between each red data point and the average line it will add up to zero. Obviously, the same is true for the blue data points! This is epressed mathematically by the equation 0 "Definition of an Average" We can obtain a measure of the spread of the data by summing the square of the distance between each data point and the average line. Since taking additional data points will increase the sum even though the data points might be closer to the average (less spread), we must divide by the number of data points. Thus, we are finding the average of the square of the distance between the data points and the average line. This is called the variance!
σ ( ) Since we have to calculate the epectation value to compute the variance, we find the following formula more useful for computations: σ The equation says that the variance is the average of the squares of a quantity minus the square of the average of the quantity. From dimensional analysis, you should realize that the variance doesn't have the same units as. Thus, we need to take the square root of the variance in order to obtain a quantity that can represent the spread of. This quantity is called the standard deviation! σ σ In physics, we usually refer to the standard deviation as the uncertainty as it represents the uncertainty in a measurement. We will find that uncertainty has a very important place in quantum mechanics. II. Binomial Distribution A. We often find events where there are only two possible results. For eample, a nucleus has either decayed or has not decayed; a particle is detected or not detected; a flipped coin either comes up heads or tails. If a system contains a number of identical events each of which are independent, have only two outcomes, and the probability for each outcome is constant, then the statistics of the system is described by the binomial distribution. B. Formula If the probability that a single event can occur is given by p than the probability f that events will occur out of n independent trials is given by f() = n! p (1 p) n! (n )!
The mean number of events that occur is given by = np The standard deviation is given by σ = np(1 p) Although many physical systems are described by the binomial distribution, it is often difficult if not impossible to use the distribution for actual calculations especially when the number of trials is large. For instance, a gram of Uranium contains more than 10 5 nuclei (n = 10 5 ) that can decay so the number of calculations to be performed would be enormous. For this reason, we usually approimate the binomial distribution with a continuous function. III. Gaussian Distribution A. The Gaussian distribution is one of the more important probability distributions. It finds application in a wide range of fields. It is sometimes called the normal or standard distribution or the Bell curve. It is also sometimes referred to as the "drunken walk." The Gaussian distribution is a special case of the discrete binomial distribution for large numbers of trials, n, where the probability of success, p, is not too small (see Appendi D of Rohlf). A common eample of this condition occurs when the physical quantity that is being measured depends on the "sum" of a set of large random numbers. Start The displacement of a drunk undergoing a random walk is the sum of several random steps. The drunk should have a much greater probability for small
Probability displacements where his/her individual steps cancel each other than for large displacements where more of the steps must be in the same direction! B. Formula The probability that events occur for a system with an average of a and a standard deviation of "σ" is given by: f () 1 πσ e ( a) /(σ ) C. Graph The graph for a Gaussian with a mean of 5 and a standard deviation of 6 is shown below. Gaussian Distribution 0.07 0.06 0.05 0.04 0.03 0.0 0.01 0 0 5 10 15 0 5 30 35 40 45 50 The solid green line shows that: 1) the "most probable" value (peak) is at = 5 ) the median (50% level) is at = 5 3) the mean is at = 5 The dashed red lines (inner pair) show the region where - < < +. The probability that a particle is located in this region is 0.683 (68.3%).
The dashed purple lines (outer pair) show the region where - < < +. The probability that a particle is located in this region is 0.95 (95%). IV. Poisson Distribution A. From the graph in the previous section, we see that the Gaussian distribution is symmetric about the mean. This symmetrical shape cannot describe a system in which there is a very small probability of an event occurring as in the case where one is trying to detect a rare particle such as the Higgs boson. In these cases, the peak of the distribution has to be near zero and be asymmetrical. A system described by a binomial distribution with a large number of trials, n, where the probability, p, for an individual event is so small that np is small is instead approimated by the Poisson distribution. This has great application in nuclear counting statistics as well as engineering applications like measuring small concentrations of a pollutant. B. For a system of n trials where the probability of a single event is p, the probability for having events is given by: f() = (np) e np! For the Poisson distribution, the mean and standard deviations are given by = np σ = np In sampling problems, where the probability of an event may not be known apriori, the probability for later samples is often assumed to be the same as from a previous sample. For instance, if a batch of 50,000 parts had three defective parts then the average for the net batch of 50,000 parts is assumed to be three and the probability is 610-5 for calculations as to the number of defects epected in later batches.
C. Graph The graph of a system described by a Poisson distribution with p = 0.0001 and n = 10000, shows the asymmetrical nature of the distribution. 0.4 0.35 0.3 0.5 0. 0.15 0.1 0.05 Poisson Distribution 0 0 1 3 4 5 6 7 8 9 10 One can see that for even 10,000 trials there is almost no probability for more than four events to occur. The average of the distribution is one.