Distributions in Excel

Transcription

1 Distributions in Excel

2 Functions Normal Inverse normal function Log normal Random Number Percentile functions Other distributions

3 Probability Distributions A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using letters such as X. A discrete random variable is a random variable that has values that has either a finite number of possible values or a countable number of possible values. A continuous random variable is a random variable that has an infinite number of possible values that is not countable. A probability distribution provides the possible values of the random variable and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula.

4 Requirements for a Discrete Probability Distribution Let P(X = x) denote the probability the random variable X equals x, then P(X = x) = 1 and 0 < P(X = x) < 1

5 probabilities A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value of the random variable. 0.7 Probability Distribution x probability random variable values

6 The Binomial Probability Distribution Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment provided: 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each trial, there are two mutually exclusive outcomes, success or failure. 4. The probability of success is fixed for each trial of the experiment.

7 Notation Used in the Binomial Probability Distribution There are n independent trials of the experiment Let p denote the probability of success so that 1 p is the probability of failure. Let x denote the number of successes in n independent trials of the experiment. So, 0 < x < n.

8 Binomial Probability Distribution Function The probability of obtaining x successes in n independent trials of a binomial experiment where the probability of success is p is given by P(X = x) = n C x p x (1-p) n-x x = 0, 1,..., n where p = probability of success

9 Poisson Distribution A random variable X, the number of successes in a fixed interval, follows a Poisson process provided the following conditions are met 1. The probability of two or more successes in any sufficiently small subinterval is The probability of success is the same for any two intervals of equal length. 3. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping.

10

11 Normal distribution The normal distribution closely approximates the probability distributions of a wide range of random variables. Normal distribution was originally developed to give a model for measurement errors. The dimensions of manufactured parts or the weights of food packages often follow a normal distribution. This leads to Quality control applications. Total sales or production often follow a normal distribution that leads us to a large family of applications in marketing and production management. The patterns of stock and bond prices are often modeled using the normal distribution in large computer-based financial trading models. Economic models use the normal distribution for a number of economic measures.

12 The normal distribution is described by the bell-shaped curve. The normal distribution is symmetric and has the property that the median equals the mean.

13

14 Who and Why Should Use Distributions? Random factors affect all areas of our life, and businesses striving to succeed in today's highly competitive environment need a tool to deal with risk and uncertainty involved. Using probability distributions is a scientific way of dealing with uncertainty and making informed business decisions. In practice, probability distributions are applied in such diverse fields as actuarial science and insurance, risk analysis, investment, market research, business and economic research, customer support, mining, reliability engineering, chemical engineering, hydrology, image processing, physics, medicine, sociology, demography etc.

15 Examples: 1. Pricing rain insurance An arranger of a sports event like to take a rain insurance. Pricing must be based on weather statistics and agreed indemnities. rain (mm) prob. indemnity % % % % 3000 Expected value of indemnity 360 Insurance price is expected value + insurance company's premium.

16 2. Investment profitability euros/year probability Investment cost Return if boom % Return if recession % Expected value of revenue Revenue percent 38% 3. Lottery tickects Amount of tickets 1000 euros count Winning tickets Expected win / ticket 5.5 euros

17 4. Warranty repair effect on car price An Importer of a car model has history data on warranty repairs: Average repair under warranty prob % % 600 5% What is the effect of warranty to the car price? Expected repair under warranty 160 euros

18 Thank you