Lecture 6. Probability events. Definition 1. The sample space, S, of a. probability experiment is the collection of all

Transcription

1 Lecture 6 1 Lecture 6 Probability events Definition 1. The sample space, S, of a probability experiment is the collection of all possible outcomes of an experiment. One such outcome is called a simple event. An event is a collection of several outcomes. The events are denoted by capital letters, and you could think of them as being sets. Definition 2. The probability of an event denoted P (E) is the likelihood of that event occurring.

2 Lecture 6 2 The probability of an event has to satisfy the following statements: 1) 0 P (E) 1. 2) If an event is impossible, the probability of the event is 0. 3) If an event is a certainty the probability of the event is 1. 4) If S = {e 1, e 2,, e n }, then P (e 1 ) + P (e 2 ) + + P (e n ) = 1.

3 Lecture 6 3 Definition 3. Let E and F be two events. E and F (E F ) is the event consisting of simple events that belong to both E and F. If the events E and F do not have any simple events in common then we say that they are disjoint or mutually disjoint. E or F (E F ) is the event consisting of simple events that belong to either E or F or both. Definition 4. Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted E, consists of all simple events in the sample space S that are not simple events in the event E.

4 Lecture 6 4 Proposition 5. a) If the events E, F, G,... are mutually disjoint then P (E or F or G or ) = P (E)+P (F )+P (G)+ b) For any two events E and F P (E or F ) = P (E) + P (F ) P (E and F ) c) For any event E, if E is its complement, we have P (E) = 1 P (E).

5 Lecture 6 5 How do we compute probabilities The classical method of computing probabilities requires equally likely outcomes. An experiment has equally likely outcomes if all simple events have the same probability of occurring. Proposition 6. If an experiment has equally likely simple events, then the probability of an event E occurring is computed by: P (E) = Number of ways that E can occur Number of possible outcomes

6 Lecture 6 6 Sometimes it is difficult or impossible to count the number of ways an event E could occur. In this case we could determine the probability based upon the outcomes of a probability experiment. The probability of an event E is approximately the number of times the event E is observed, divided by the number of repetitions of the experiment. P (E) = relative frequency = frequency of E number of trials of experiment.

7 Lecture 6 7 Probability events Definition 7. If E and F are any two events, then the probability of event E given the event F (or theconditional probability of E given F is computed by: P (E F ) = P (E and F ). P (E) The probability of the event F occurring given the occurance of event E is found by dividing the probability of E and F by the probability of E. As a consequence we have the general multiplication rule: P (E and F ) = P (E)P (E F ).

8 Lecture 6 8 Definition 8. Two events E and F are independent if the occurance of event E in a probability experiment does not affect the probability of event F. In mathematical notations, two events are independent if P (E F ) = P (E) or P (F E) = P (E) If the events are not independent we say they are dependent. Multiplication rule for 2 independent events Two events are independent if and only if (iff) P (E and F ) = P (E)P (F ). This means that if two events are independent then it must be the case that P (E and F ) = P (E)P (F ) and if P (E and F ) = P (E)P (F ) holds then the events

9 Lecture 6 9 E and F are independent.

10 Lecture 6 10 Multiplication rule for n independent events If events E, F, G are independent, then P (E and F and G ) = P (E)P (F )P (G)

11 Lecture 6 11 Counting techniques Multiplication principle If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the task of making these selections can be done in p q r different ways.

12 Lecture 6 12 Number of Combinations of n distinct objects taken r at a time The number of different arrangements of n objects using r n of them, in which 1) the n objects are distinct 2) once an object is used it can not be repeated (without replacement) 3) order is not important is given by the formula C n r = n! r!(n r)!

13 Lecture 6 13 Probability Distributions In certain situations, some attribute of the outcome may hold more interest for the experimenter than the outcome itself. For example, a player of the game of craps may be concern only about throwing a 7 and not weather the 7 was the result of a 5 and a 2 or a 4 and a 3 or a 6 and a 1. Definition 9. A random variable (r.v.) is a numerical measure of the outcome of a probability experiment, so its value is determined by chance. Random variables are denoted using capital letters such as X, Y, etc.

14 Lecture 6 14 Definition 10. A discrete random variable is a random variable 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.

15 Lecture 6 15 Because the value of a r.v. is determined by chance, there are probabilities assigned to these possible values. A table, graph, or formula containing all the possible values a random variable could take together with the corresponding probabilities forms a probability distribution. In the case of a discrete probability distribution the following equalities must be verified: 1) P (X = x) = 1 2) 0 P (X = x) 1 where (X = x) denotes the probability of the

16 Lecture 6 16 random variable X to be x.

17 Lecture 6 17 Mean and variance of a discrete random variable The mean, or the expected value, of a discrete random variable is given by the formula µ X = E(X) = x P (X = x) where x is the value of the random variable and P (X = x) is the probability of observing the random variable x.the variance of a discrete r.v. is given by σ 2 X = (x µ X ) 2 P (X = x) and the standard deviation is the square root of the variance, i.e. σ X = σ 2 X.

18 Lecture 6 18 Binomial distribution When do we deal with a binomial trial or distribution? An experiment is said to be a binomial experiment if: 1) The experiment is performed a fixed number of times, usually denoted by n. Each repetition is called a trial. 2) The trials are independent (the outcome of one does not depend on the other) 3) For each trial, there are 2 mutually exclusive outcomes: success or failure. 4) The probability of success is fixed for each trial of the experiment.the probability of success is p while of failure is 1 p 5) We say that a r.v. is binomially distributed if X counts the number of

19 Lecture 6 19 successes in n independent trials of the experiment. So the possible values for X are 0, 1, 2,..., n.

20 Lecture 6 20 Mathematicians showed that 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, 2,..., n Also they showed that such a binomial random variable will have the mean given by µ X = E(X) = np and the standard deviation given by the formula: σ X = np(1 p)

21 Lecture 6 21 Continuous r.v. s Normal Distribution In the case of continuous r.v. s, computing probabilities is not that easy because the r.v. takes infinitely many values. That is why we look at intervals of values the r.v. might take. Probability density Function A probability density function is a function used to compute probabilities of continuous r.v. s. It has to satisfy the following two properties: (1.) The area under the graph of the equation over all possible values of the r.v. must equal one.

22 Lecture 6 22 (2.) The graph of the equation must lie on or above the x-axis for all possible values of the r.v.

23 Lecture 6 23 Property: The probability of observing a value of the r.v. in a certain interval equals the area under the graph of the density function of that r.v., over that interval. A continuous r.v. is normally distributed or has a normal probability distribution if its relative frequency histogram has the shape of a normal curve (bell-shaped and symmetric).

24 Lecture 6 24 Area and the normal distribution If the r.v. X is normally distributed then the area under the normal curve for any range of values of the r.v. X represents either: 1) the proportion of the population with the characteristics described by the range, or 2) the probability that a randomly chosen individual from the population will have the characteristics described by the range.

25 Lecture 6 25 Finding the area under the density graph of a normally distributed r.v. is not an easy task. It requires a lot of calculus. One way of avoiding this is to use tables that give us these areas (probabilities). But for each µ and σ we would need a new table. How can we avoid this? By transforming somehow all these r.v. into a standard one. Standardizing a normal r.v. Suppose that the r.v. X is normally distributed with mean µ and standard deviation σ. Then the r.v. Z = X µ σ

26 Lecture 6 26 is normally distributed with mean µ = 0 and standard deviation σ = 1 Such an r.v. is said to have the standard normal distribution.