(ii) Poisson distribution

1. (ii) Poisson distribution • The Poisson distribution resembles the binomial distribution if the probability of an accident is very small. • Some events are rather rare, they don't happen that often. Still, over a period of time, we can say something about the nature of rare events. • The Poisson distribution is most commonly used to model the number of random occurrences of some phenomenon in a specified unit of space or time. CY1B2 Statistics

2. (ii) Poisson distribution Let p become very small, n very large, but finite. then which depends on r only Mean value Variance The Poisson distribution applies to many physical phenomena, such as radioactive decay, in which n (the number of atoms) is very large but p (the probability of a given atom decaying in a given time) is very low. … CY1B2 Statistics

3. 6 Example: Over a given period there are 10 cars on the road, and the probability of a given car having an accident is 10 . Find the mean number of accidents , and variance. -4 Mean value Variance The standard deviation is then 10. This means a 10% rate in the accident rate over a given period is not significant. CY1B2 Statistics

4. Discrete Distribution: A statistical distribution whose variables can take on only discrete values. • The mathematical definition of a discrete probability function, p(x), must satisfies the following properties. • The probability that x can take a specific value is p(x). That is • p(x) is non-negative for all real x. • The sum of p(x) over all possible values of x is 1. CY1B2 Statistics

5. Continuous Distribution: A statistical distribution for which the variables may take on a continuous range of values. The mathematical definition of a continuous probability function, f(x), satisfies the following properties: • The probability that x is between two points a and b is • It is non-negative for all real x. • The integral of the probability function is one, that is CY1B2 Statistics

6. What does this actually mean? Since continuous probability functions are defined for an infinite number of points over a continuous interval, the area under the curve between two distinct points defines the probability for that interval. (Probabilities are measured over intervals, not single points.The probability at a single point is always zero.) The height of the probability function can in fact be greater than one. The property that the integral must equal one is equivalent to the property for discrete distributions that the sum of all the probabilities must equal one. CY1B2 Statistics

7. Continuous probability distributions (i) Probability density Suppose we sample a waveform as shown in above figure, with a digital voltmeter, taking a reading with a resolution δx (e.g. =0.1V), over a period of time t. CY1B2 Statistics

8. In order to intuitively develop the concept of continuous probability function, we uses the method of histogram, and then see how to define continuous probability function from histogram. • A histogram is obtained by splitting the range of the data into equal-sized bins (called classes). Then for each bin, the number of points from the data set that falls into each bin is counted. That is • Vertical axis: Frequency (i.e., counts for each bin) • Horizontal axis: Response variable • The histogram has an additional variant whereby the counts are replaced by the normalized counts. The names for these variants are the relative histogram and the relative cumulative histogram. CY1B2 Statistics

9. A histogram as shown above gives the number of reading n versus range x. The right hand side plots versus x, where n is the total number of reading, when n becomes very large. CY1B2 Statistics

10. The first step normalized count is the count in a class divided by the total number of observations. In this case the relative counts are normalized to sum to one (or 100 if a percentage scale is used). This is the intuitive case where the height of the histogram bar represents the proportion of the data in each class. Or the probability of the data falling into each range (each class). Sums up to one. CY1B2 Statistics

11. To get a smoother curve, reduce δx - 0.1v,0.01v,0.001v, 0.0001v- 0, such that each range becomes infinitely small, but the problem is that data points in every class nx/n 0, becomes negligible or  0. 2. The second step normalized count is the count in the class divided by the number of observations times the class width. For this normalization, the area (or integral) under the histogram is equal to one. The area under the curve is unity. CY1B2 Statistics

12. The resultant smooth curve is a plot of probability density function p(x) CY1B2 Statistics

13. Clearly p(x)δx is the probability of a reading being in a small range δx . The probability of a reading between two values The area under the curve is unity. CY1B2 Statistics

14. The expressions for the mean, mean square and variance for continuous probability distribution p(x) are The mode: value of x that corresponds to maximum p(x) The Median: value of x that divides plot into equal areas. CY1B2 Statistics

15. Example: Deduce the probability density for the sawtooth waveform. CY1B2 Statistics

16. It can be deducted that the wave form moves uniformly through all range of [0,A]. CY1B2 Statistics