Chapter 1 - Probability Distribution PDF

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Warning: TT: undefined function: 32 STA408: Statistics for Science and EngineeringChapter 1: Probability Distribution1 ProbabilitySample Space The set of all possible outcomes of a statistical experiment is called a sample space and is represented by the symbol S.Example 1: Consider tossing a fair d...

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STA408: Statistics for Science and Engineering

Chapter 1: Probability Distribution 1.1

Probability

Sample Space The set of all possible outcomes of a statistical experiment is called a sample space and is represented by the symbol S. Example 1:

Consider tossing a fair die. If we are interested in the number that shows on the top face, the sample space is S  {1, 2, 3, 4, 5, 6}. If we are interested in whether the number is odd or even, the sample space will be S  {odd, even}.

Events An event is a subset of a sample space. Example 2:

Consider tossing a fair die. Let event A be the outcome that when the die is tossed, the number shows on the top face is divisible by 3. A  {3, 6}

Complement The complement of an event A with respect to S is the subset of all elements in S that are not in A. We denote the complement of A by the symbol A. Example 3:

Consider the event in Example 2. The complement of event A is A  {1, 2, 4, 5}.

Probability of an Event The probability of an event A is the sum of all weights/probabilities of all sample points in A and is denoted by 𝑃(𝐴). Therefore and 0≤𝑃(𝐴) ≤1, 𝑃(∅) =0, 𝑃(𝑆) =1. If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A is 𝑛 𝑃(𝐴) = . 𝑁 If A and A are complementary events, then

𝑃(𝐴) +𝑃(𝐴′) =1.

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Chapter 1: Probability Distribution

Random Variables

Concept of a Random Variable A random variable is a function that associates a real number with each element in the sample space. Example 4:

The sample space giving a detailed description of each possible outcome when three electronic components are tested may be written as S  {NNN, NND, NDN, DNN, NDD, DND, DDN, DDD} where N denotes non-defective and D denotes defective. Let the random variable X be the number of defective items when three electronic components are tested, then Sample Space

𝑥

NNN

0

NND or NDN or DNN

1

NDD or DND or DDN

2

DDD

3

Discrete Sample Space If a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers, it is called a discrete sample space. A random variable is called a discrete random variable if its set of possible outcomes are countable. Example 5:

Some discrete random variables: - Number of defective items - Number of calls in a minute - Number of accidents on the highway

Continuous Sample Space If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space. When a random variable can take on values on a continuous scale, it is called a continuous random variable. In most practical problems, continuous random variables represent measured data. Example 6:

Some continuous random variables: - Heights - Weights - Time - Distance

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Chapter 1: Probability Distribution

Special Discrete Probability Distribution

Binomial Distribution A binomial experiment is a probability experiment that satisfies the following four requirements.    

There must be a fixed number of trials. Each trial can have only two (2) outcomes or the outcomes that can be reduced to two (2). These outcomes can be considered as either a success or a failure. The outcomes of each trial must be independent of one another. The probability of success must remain the same for each trial.

The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution. Notation for symbols used in a Binomial Distribution 𝒑

Probability of a success

𝒒 or 𝟏−𝒑

Probability of a failure

𝒏

Number of trials

𝒙

The number of successes in 𝑛 trials.

Note: 𝑥 =0, 1, 2, … , 𝑛; and 𝑝+𝑞 = 1. Example 7 Ten percent of all DVD players manufactured by a large electronic company are defective. Five DVD players are randomly selected from the production line of this company. The selected DVD players are inspected to determine whether each of them is defective or good. Is this experiment a binomial experiment?  There are _______________ DVD players (_______________ trials); 

___________________outcomes, i.e., ______________________or _______________________ ;



Each DVD player is ______________________________________ of each other;



The probability of a defective DVD player is __________________, i.e., 𝑝= _________________.

Since the four conditions of a binomial experiment are satisfied this is an example of a binomial experiment. Binomial Probability Formula In a binomial experiment, the probability of exactly 𝑋 successes in 𝑛 trials is 𝑛 𝑃(𝑋=𝑥) =( ) 𝑝 𝑥 (1− 𝑝)𝑛−𝑥 ; 𝑥 = 0, 1, 2, … , 𝑛 . 𝑥 Note that 𝑋 is a discrete random variable. If a random variable 𝑋 has a binomial distribution with 𝑛 trials and the probability of success of each trial is given as 𝑝, then 𝑋 can be written as 𝑋 ~ 𝐵𝑖𝑛(𝑛, 𝑝)

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Chapter 1: Probability Distribution

Example 8 Refer to Example 7, what is the probability that exactly one of these five DVD players is defective?

Example 9 A fair coin is tossed 3 times. Find the probability of getting exactly two heads.

Example 10 A bag contains a large number of beads of which 45% are yellow. A random sample of 10 beads is taken from the bag. Use a binomial distribution to calculate the probability that the sample contains (a) more than 6 yellow beads; (b) exactly 6 yellow beads; (c) at most 6 yellow beads; (d) less than 6 yellow beads; (e) at most 6 beads which are not yellow.

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Chapter 1: Probability Distribution

The Mean (Expected Value), Variance and Standard Deviation of a Binomial Distribution Mean, 𝜇 Variance, 𝜎 2 Standard Deviation, 𝜎

𝑛𝑝 𝑛𝑝𝑞 =𝑛𝑝(1−𝑝) √𝑛𝑝𝑞= √𝑛𝑝(1−𝑝)

Example 11 A coin is tossed 3 times. Find the mean, variance and standard deviation for the number of tails that will be obtained.

Example 12 A die is rolled 350 times. Find the expected value, variance and standard deviation for the number of fours that will be rolled.

Poisson Distribution A discrete probability distribution that is useful when  

Sample size, 𝒏 is large and probability, p is small. The independent variables occur over a period of time, or when a density of items is distributed over a given area / volume. 5

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Chapter 1: Probability Distribution

Poisson Probability Formula The probability of 𝑋 occurrences in an interval of time, volume, area, etc., for a variable where 𝜆 is the mean number of occurrences per unit (time, volume, area, etc.) is 𝑒 −𝜆 𝜆𝑥 𝑃(𝑋 =𝑥) = ; 𝑥 = 0, 1, 2, … 𝑥! where 𝑒 is a constant 2.718281 … . If a random variable 𝑋 has a Poisson distribution with a mean 𝜆, then 𝑋 can be written as 𝑋 ~ 𝑃𝑜(𝜆) Example 13 During a laboratory experiment, the average radioactive particles passing through a counter in 1 millisecond is 4. What is the probability that in a given millisecond (a) no particles enter the counter, (b) at least 4 particles enter the counter, (c) more than 4 particles enter the counter (d) at most 4 particles enter the counter.

Example 14 The number of accidents per week at a certain road intersection has a Poisson distribution with parameter 2.5. Find the probability that (a) exactly 5 accidents will occur in a week; (b) at most 5 accidents will occur in two weeks; (c) less than 5 accidents will occur in four weeks; (d) more than 1 accident will occur in three weeks.

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Example 15 Eight is the average number of oil tankers arriving each day at a certain port. The facilities at the port can handle at most 15 tankers per day. (a) What is the probability that on a given day tankers have to be turned away. (b) Assume that there are 24 hours in a day. What is the probability that there is no tanker arriving in 6 hours?

Mean (Expected value), Variance and Standard Deviation for a Poisson Distribution Mean, 𝜇

𝜆

Variance, 𝜎 2

𝜆

Standard Deviation, 𝜎

√𝜆

Example 16 Refer to Example 15 where the number of accidents per week at a certain road intersection has a Poisson distribution with parameter 2.5. Let 𝑋 be the number of accidents per week, find the expected number of accidents per week. Find also the variance and standard deviation for the distribution of 𝑋.

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Chapter 1: Probability Distribution

Probability of Success and the Shape of the Binomial Distribution For any number of trials n:  The binomial probability distribution is symmetric if 𝑝=0.5.  The binomial probability distribution is skewed to the right if 𝑝0.5. Probability distribution of 𝑥 for 𝑛=4 and 𝑝=0.5 𝑥

𝑃(𝑥)

0

0.0625

1

0.2500

2

0.3750

3

0.2500

4

0.0625

Probability distribution of 𝑥 for 𝑛=4 and 𝑝=0.3 𝑥

𝑃(𝑥)

0

0.2401

1

0.4116

2

0.2646

3

0.0756

4

0.0081

Probability distribution of 𝑥 for 𝑛=4 and 𝑝=0.8 𝑥

𝑃(𝑥)

0

0.0016

1

0.0256

2

0.1536

3

0.4096

4

0.4096

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Chapter 1: Probability Distribution

Mean and the Shape of the Poisson Distribution The form of Poisson distribution becomes more and more symmetric, even bell-shaped, as the mean grows large. The figure below shows the plots of probability function for 𝜇=0.1, 𝜇 =2 and 𝜇=5.

Poisson Distribution as an Approximation to Binomial Distribution In the case of binomial, if 𝑛 is quite large and 𝑝 is small, the conditions begin to simulate the continuous space or time implications of the Poisson process. Therefore,  If 𝑛 is large and 𝑝 is close to 0, the Poisson distribution can be used, with 𝜇 =𝑛𝑝, to approximate binomial probabilities.  If 𝑝 is close to 1, can still use the Poisson distribution to approximate binomial probabilities by interchanging what we have defined to be a success and a failure, i.e., by changing 𝑝 to a value close to 0. Let 𝑋 be a binomial random variable with probability distribution 𝐵𝑖𝑛(𝑛, 𝑝). When 𝑛 →∞, 𝑝→0 and 𝑛→∞

𝑛𝑝 →

𝜇 remains constant,

𝑛→∞

𝐵𝑖𝑛(𝑛, 𝑝) →

𝑃𝑜(𝜇)

Example 17 In a certain industrial facility, accidents occur frequently. It is known that the probability of an accident on any given day is 0.005 and accidents are independent of each other. What is the probability that any given period of 400 days there will be an accident on one day?

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Chapter 1: Probability Distribution

Continuous Probability Distribution

Probability of a continuous random variable 𝑃(−1≤𝑥 ≤2) = area bounded by the curve, 𝑥-axis, 𝑥 =−1 and 𝑥=2.

The probability of a continuous random variable 𝑥 between the values 𝑎 and 𝑏 is the area under the curve and between the vertical lines 𝑥 =𝑎 and 𝑥 =𝑏 as given below. 𝑃(𝑎 ≤𝑥 ≤𝑏) =𝑃(𝑎...