PROBABILITY & STATİSTİCS

Prof. Dr. Orhan TORKULRes. Assist. Furkan YENER

Probability & Statistics Many of problems we face daily as industiral engineers have

elements of risk, uncertanity, or variabilty associated with them. For example, we cannot always predict what the demand will be for a particular inventory item. We cannot always be sure just how many people will shop at a grocery store and desire to check out during a particular hour. We cannot always be sure that the quality of raw materials from one of our suppliers will be consistent. We are not prophets who can accurately predict results in advance. But we as industrial engineers are educated in the use of applied probability and statistics to make intelligent engineering decisions despite our lack of complete knowledge about future events.

Probability of an Event An event may consist of any combination of possiable outcomes of

an experiment. It is up to the experimenter to define events that are meaningful in the experiment. Let us consider the experiment of drawing one ball from a box containing ten balls, numbered 6 through 10 are white. The probability of drawing any particular one of the ten balls on any performance of the experiment is 0,1. We can define many events relating to the experiment of drawing one ball from the box as follows:

is the event of drawing an even-numbered ball is the event of drawing green ball is the event of drawing even-numbered green ball.

Sample Points Counting Rules Addition rule: For two seperated event; firtst one has different

possibilities, second has different possibilities. One of these events may happen + different probablity.

Ex: If we throw a dice, what is the possiblities for odd and even numbers? Solution: There are 3 odd and 3 even number on a dice so that 3 + 3 = 6 is the number of possibilities.

Multiplication Rule: For two non-seperated event; firtst one has different possibilities, second has different possibilities. One of these events may happen x different probablity.

Ex: If a coin has tossed 3 times, what is the number of probabilities?Solution: when a coin has tossed, there are 2 possibilities. If it happens 3 times, 2 x 2 x 2 = 8 different event can arise.

Combinations In considering problems involving finite sample spaces of equally

likely outcomes, we are often tempted to count the fruquencies of interest. This procedure is perfectly valid, but counting may be time consuming and tedious. Therefore, we most often use a formula to determine numbers of combinations. The number of combinations of n things taken k at a time is expressed by the following notation and formula:

Estimating Probabilites Let us consider an experiment to be the inspection of a reel of magnetic

tape produced by a certain process. Each repetition of this experiment will consist of the examinations of a different reel of tape, the possible decisions being accepti rework and reject. The number of repetitions corresponds to the number of reels of tape, and the possible decisions number only three; thus, both the number of repetitions and number of decisions are finite.

number of repetitions of the experiment that will result in decision number of different possible decisions total number of repetitions probability to happen .

()

Some Important Probability Distributions1. Discrete Distribution Properties2. Binomial Distribution3. Poisson Distribution4. Discrete Uniform Distribution5. Uniform Distribution6. Normal Distribution7. Exponential Distribution8. Rectangular Distribution

Normal Distributions We shall define the general form of the normal distrubition for the continuous

random variable as follows: for ;This distribution is one of the most interesting and useful that we can study. It has a single peak at the mean and is symmetrical about that point. If we plot an example of a normal distribution, it will be readily apparent that it is bell-shaped with mean µ and variance . In practice, many distributions are well approximated by the normal distribution. Some examples include bolt diameter, construction errors, resistance of a specified type of wire, weight of a packaged material, and so on.

Expected Values and Variability Mean;The mean or expected value of discrete random variable x is denoted by the letter and is defined as follows:

Where is the probability that takes on the value .

Variance;The measure of variability that we have considered is called the variance(), and it is defined as follows.

Populations and Samples Much of the work of the applied professions involves the study of

only a subset of the total items of interest, in the hope of making statistical inferences about the total. An engineer might collect data on machine utilization for 1 month, hoping to infer from it machines utilization information for many months or years. An automobile manufacturer might test small number of automobiles and then make generalized statements about all the automobiles produced during that model year. An inspection team might use destructive inspection on a small percentage of items in order to infer caracteristics of the total number beng produced. In order to describe this process accurately, we must clearly understand the meaning of population and sample.

Population A population, in the broadest sense, is the total set of elements

about which knowledge is desired. Some populations are relatively small, for example, the number of space shuttles; other populations are large, for example, all the electric light bulbs now in existence and to be produced in the future. All elements of a population do not have to be in existence, as the last example indicates. The important thing to remember is that the population must be definable.

Sample A sample is a subset of a population. In extreme situations the

sample may be the complete population or it may consist of no elements at all. Of course, this latter sample would yield no information and we shall not consider it further. Remember that the purpose of a sample is to yield inferences about the population from which it was taken.

The two most important futures of a sample are its size and the manner in which it was selected. Much of the study of sampling statistics concerns the determination of these two characteristics. As expected, this determination is based on the specific conditions prescribing the purpose of the sample.

Sample Statistics A sample statistic is a value calculated from a

sample that may be used to estimate a population parameter such as a mean or variance.Two important sample statistics are the sample mean and the sample variance.

The sample mean is defined as follows:

The sample variance is defined as follows:

Distribution of Sample Means We often make inferences about a population from the average

value of a sample.This usually requires that we know the parameters of the distribution of means. Naturally, the expexted value of the sample average is µ, the same mean value as held by the population. The variance of the sample means , differs from the population variance and is given by the following.

EXAMPLE: The delay times (handling, setting, and positioning the tools) for cutting 6 parts on an engine lathe are 0.6 1.2 0.9 1.0 0.6 and 0.8 minutes. Calculate variance.Solution: First we calculate the mean: = = 0.85 Then we set up the work required to find in the following table:

We divide 0.2750 by 6−1 = 5 to obtain == 0.055 = Variance.

-0,6 -0,25 0,06251,2 0,35 0,12250,9 0,05 0,00251 0,15 0,0225

0,6 -0,25 0,06250,8 -0,05 0,00255,1 0,00 0,2750

Example

Central Limit Theorem In essence, Central limit theorem says: ıf has a distribution with a

finite variance , then the random variable has a distribution that approaches normality as the sample size tends to infinity. Fortunately, for many population distributions often encountered, sample sizes as low as produce sample average distributions which are workably close to normal.

We use the central limit theorem extensively in quality control, probabilistic models, or project manager.

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