 Welcome to dealing with materials data. In this course we are looking at collection analysis and interpretation of data from material science and engineering. We are looking at probability distributions and in this session we are going to look at lifetime and exponential distributions because they are also important in many material science and engineering problems. Suppose what is lifetime? Suppose let us consider a short laser pulse that excites a fluorescent molecule at time t equal to 0. What is the lifetime of the molecule in the excited state is a question that you can ask. So it is going to stay in the excited state for a while and then it is going to come back by emitting the fluorescent light. Now f of t is the probability density function which gives the normalized time dependent intensity of radiation or light. Suppose if you have large number of molecules that are excited at time t equal to 0, in a short time delta t how many molecules will emit light radiation? Let us say between t and t plus delta t. So that is given by f of t times delta t. So f of t is the probability density function or normalized time dependent intensity function. Then f of t which is the cumulative distribution function of the normalized time dependent intensity basically gives the fraction with lifespan less than or equal to t and 1 minus f of t which is called the survival function gives the fraction that survives at time t. So we are saying that at time t equal to 0 we excite a large number of molecules and then by the time you reach time t what is the fraction which would have come back by emitting light to their from the excited state and how many survive in the excited state. So this is given by f of t and 1 minus f of t. We can also define what is known as a hazard function. It is a failure rate function and it is defined as f of t divided by 1 minus capital f of t. f of t is a derivative of this function because this is obtained by integrating f of t. So this is, so the f of t is basically derivative of this function. So you can integrate, so you will get f of t to be h of t exponential minus integral 0 to t h t prime dt prime. So in terms of the hazard function or failure rate function you can write the probability distribution function and this is the relationship and that happens to be exponential. Now the simplest choice that you can make for the hazard rate function is that it is a constant k and what does the k the rate constant tell you it tells you the relative fraction of members of a population that disappear in unit time. So if you had n excited molecules and dn by dt are the ones which will emit radiation in unit time. So dn by dt by n is basically k which is the rate constant it is a relative fraction of members of a population that disappear in unit time. So this kind of description is for first order chemical reactions and radioactive decay of materials. So all this follow this dn by dt is minus kn which means that they will be described by the so called exponential distribution k exponential minus kt and the f of t is 1 minus exponential minus kt. So this is the exponential distribution and you can also go back to the nucleation problem that we looked at. So just to remind you so this is the exponential distribution and f of t is 1 minus exponential minus kt. In the nucleation problem we showed that it is a Poisson distribution and P of m so for m nucleate to form we said that it is n power m by m factorial exponential minus n. So if you calculate P of 0 that is probability that there is no nucleation in a given time t. Remember the t was part of this n then you get exponential minus n and if you ask for at least one nucleate to have formed in time t that will be given by 1 minus exponential minus n. So you can see that experimental data on crystal nucleation rates and induction times for example follow this distribution which is the exponential distribution and one example is this crystal nucleation rates from probability distributions of induction times it is crystal growth and design and there is data that is given and you can look at the numbers and analyze and look at how these distribution functions look like. Weibull distribution is a generalized exponential distribution we have already worked with Weibull distribution and in this case the hazard function is C t power C minus 1 and if C is 1 it is the exponential distribution if C is greater than 1 you expect higher failure rates at later times and if C is less than 1 you expect higher failure rates at early times and the probability distribution function is C t c power minus 1 exponential minus t to the power c and you can include additional parameters such as translation and scaling of t. Weibull distribution is very important in materials because failure for example many a times is described using Weibull distribution we will work with some data and show that you can describe the failure using the Weibull distribution function. Of course you can use exponential and Weibull using R, exp is the one for exponential so dxp, pxp, q exp and rexp will work and Weibull we have already used and dpqr is for the Weibull distribution function.