probability for beginners masters in africa.ppt

With illustration

1. Probability Rules And Theoretical Distributions Dr. Karuna M Dept. of Public Health Science K I University

2. Introduction We use data from a sample to draw conclusions about the population from which it is drawn The theory of probability enables us to link samples and populations, and to draw conclusions about populations from samples.

3. Probability Most aspects of life involve probabilities and medicine is no exception What is the probability that a patient will respond to a particular treatment What is a probability that a patient’s caregiver develops depression. Given appropriate data, statistical methods help to answer many such questions.

4. What is Probability? The foundation upon which the logic of inference is built Quantitative measure of uncertainty Measure of the degree of chance or the likelihood of occurrence of an uncertain event Measured by a number between 0 and 1

5. Basic Definitions Experiment: Any process of observation or measurement or any process which generates a well defined outcome Probability/Random experiment: An experiment that can be repeated any number of times under similar conditions and it is possible to enumerate the total number of outcomes without predicting an individual outcome

6. Basic Definitions… Outcome: The result of a trial of a single experiment Trial: Particular performance of a random experiment Sample space: Set of all possible outcomes of a probability experiment Event: It is a subset of sample space. It is denoted by capital letter. Equally likely events: Events which have the same chance of occurring. Complement of an event: Non occurrence of A, denoted by A’ or A c .

7. Elementary event: an event having only a single element or sample point. Mutually exclusive events: Two events which cannot happen at the same time. Independent events: Two events are independent if the occurrence of one does not affect the probability of the other occurring. Dependent events: Two events are dependent if the first event affects the outcome or occurrence of the second event in a way the probability is changed.

8. Measuring Probability All outcomes are equally likely Total number of outcomes is finite, say “N” Definition: If a random experiment with N equally likely outcomes is conducted and out of these N A outcomes are favorable to the event A, then the probability that event A occur denoted P(A) is defined as:

9. Example2: A box of 20 candles consists of 8 defective & 12 non-defective candles. If 3 candles are selected at random, What is the probability that (a) all the three candles are non-defective. (b) two are non-defective. Solution: Let A be the event that all the three are non-defective

10. Axioms of Probability Let E be a random experiment and S be a sample space associated with E. With each event A, the probability of A satisfies 3. If A and B are mutually exclusive events, the probability that one or the other occur equals the sum of the two probabilities 4. 5. 6.

11. Dependent and Independent events

12. Conditional probability of an event

13. Random Variable(R.V)

14. Types of R.V Discrete variables: which can assume only a specific number of values. They have values that can be counted. Number of children in a family Number of bacteria per cubic centimeter of water Number of car accidents per week Continuous variables: which can assume all values between any two values. Height of students at certain university Life time of light bulbs Mark of students in a class

15. Probability Distribution

16. Properties of a prob. Distn .

17. Expectation

18. Mean and variance of a random variable

19. Find the mean and variance of a r.v X in the following probability distribution

20. Binomial Distribution Assumptions of a binomial distribution The experiment consists of n identical trials. Each trial has only one of the two possible mutually exclusive outcomes, a success or a failure. The probability of each outcome does not change from trial to trial. The trials are independent, thus we must sample with replacement.

21. Examples of binomial experiments Tossing a coin 20 times to see how many tails occur. Asking 200 people if they watch BBC news. Registering a newly produced product as defective or non defective. Rolling a die to see if a 5 appears.

22. S uppose that an examination consists of six true and false questions, and assume that a student has no knowledge of the subject matter. The probability that the student will guess the correct answer to the first question is 30%. Likewise, the probability of guessing each of the remaining questions correctly is also 30%. a) What is the probability of getting exactly 3 correct answers? b) What is the probability of getting exactly 5 correct answers?

23. Solution Let X = the number of correct answers that the student gets. X ~ Bin ( n = 6, p = 0.30)  a) b)

24. Poisson Distribution

25. If X is a binomial random variable with parameters n and p then E(X) = np , Var ( X ) = npq If X is a Poisson random variable with parameters λ then E (X ) = λ , Var (X ) = λ

26. Normal Distribution Graph of the Normal Probability Density Function  f ( x ) x

27. Normal Curve The shape of the normal curve is often illustrated as a bell-shaped curve. The highest point on the normal curve is at the mean of the distribution. The normal curve is symmetric. The standard deviation determines the width of the curve.

28. Normal Curve The total area under the curve the same as any other probability distribution is 1. The probability of the normal random variable assuming a specific value the same as any other continuous probability distribution is 0. Probabilities for the normal random variable are given by areas under the curve.

29. The Normal Probability Density Function where  = mean  = standard deviation  = 3.14159 e = 2.71828

30. Importance of Normal Distribution Most of the biometric characters tend to follow normal distributions All distributions can be converted into a normal distribution When sample size is large the sampling distributions will become normal .

31. Normal distribution with varying parameters

32. Properties of Normal Distribution Bell Shaped and symmetrical about its mean Tails of the curve never touches X axis (asymptotic) It is easy to convert any distribution with any mean and standard deviation into distribution with mean 0 and SD 1 . Area Property Mean 1S.D => 68.26% Mean  2S.D=> 95.46% Mean  3S.D=> 99.74% Based on the Normal Distribution the test of significance can be constructed. Mean=Median=Mode

33. Properties … It is a continuous distribution Total area under the curve is 1, i.e., the area of the distribution on each side of the mean is 0.5. It is unimodal , i.e., values mound up only in the centre of the curve. The normal distribution is completely described by two parameters: mean and standard deviation The probability that a random variable will have a value between any two points is equal to the area under the curve between those points.

34. A random variable that has a normal distribution with a mean of zero and a standard deviation of one is said to have a standard normal probability distribution . The letter z is commonly used to designate this normal random variable. The following expression convert any Normal Distribution into the Standard Normal Distribution Standard Normal Probability Distribution

35. Properties of std. Normal Distribution Same as normal distribution, but Mean is zero Variance is one Standard deviation is one Area under the standard normal curve have been tabulated as the area between Z=0 and a positive value of Z

36. Problems in area under standard normal curve

37. Thank you