special probability distribution
TRANSCRIPT
Probability Distribution - Bernoulli - Binomial -Poisson -Normal
-GeometricS p e c i a l P r o b a b i l i t y D i s t r i b u t i o
n - B e r n o u l l i - B i n o m i a l - P o i s s o n - N o r m a
l - G e o m e t r i c Mohammad Nasir Abdullah
Senior Lecturer Department of Statistics,
Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA,
Perak Branch, Tapah Campus.
Probability Distribution?
• A statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.
• This range will be bounded between the minimum and maximum possible values, but precisely where the possible value is likely to be plotted on the probability distribution depends on a number of factors.
• These factors include the distribution's mean (average), standard deviation, skewness, and kurtosis.
Discrete Distributio
Continuous Distribution
A statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.
Bernoulli Distribution (1)
• An experiment often consists of repeated trials, each with two possible outcomes that may be labelled as “success” or “failure”.
• The Bernoulli distribution is a discrete distribution of the outcome of a single trial with only two outcomes (0 or 1), with a probability of success and probability of failure (1- ).
• Example: • Coin Tosses: record how many coins land heads up and how many land tails
up.
• Births: How many boys are born and how many girls are born each day.
• Rolling Dice: The probability of a roll of two die resulting in a double six.
5
• A single trial
• The trial can result in one of two possible outcomes, (i.e: success or failure).
• P(success) = P or theta
• P(Failure) = 1-P or 1-theta.
Quick Test!
• In a Bernoulli trial, the probability of success is 0.3. What is the probability of failure?
• The probability of success in a Bernoulli trial is 0.3. What is the variance?
8
Example (1)
• Approximately 1 in 200 Malaysian adults are lawyers. One Malaysian adult is randomly selected. What is the distribution of the number of lawyers?
• = = ( 1
Binomial Distribution
• Data often arise in the form of counts or proportions which are realizations of a discrete random variable.
• A common situation is to record how many times an event occurs in n repetitions of an experiment, i.e., for each repetition the event either occurs (a "success") or it does not (a "failure").
10
1. There are n trials.
2. Each trial results in a success or a failure.
3. The probability of a success, p, is constant from trial to trial.
4. The trials are independent.
• An experiment satisfying these four conditions is called a binomial experiment.
• The outcome of this type of experiment is the number of successes, i.e., a count.
• The discrete variable X representing the number of successes is called a binomial random variable.
• The possible counts, X = 0, 1, 2, ..., n, and their associated probabilities define the binomial distribution, denoted by X~B(n,p).
11
Example (1)
Assume that 25% of fuses are defective, and the fuses in packages of six fuses are independently selected.
a) What is the probability that (exactly) two fuses in a package of six are defective?
b) What is the probability that fewer than two are defective?
c) Find the expected number of defective fuses.
13
Example (2)
A quality control inspection system requires that from each batch of items a sample of 10 is selected and tested. If 2 or more of the sample are defective the whole batch is rejected. If the probability of an item being defective is 0.05.
a) what is the probability of 2 defectives in the sample?
b) what is the probability of the batch being rejected?
14
Example (3)
In a certain country the percentage of illiterate adults is 30%. A random sample of 10 adults is selected at random in that country. Find the probability that
a) exactly five adults are illiterate
b) at least 4 adults are illiterate
Calculate the mean and variance for the illiterate adults in that country.
15
Example (1)
In a certain manufacturing process, it is known that, on the average, 1 in every 100 items is defective. What is the probability that the fifth item inspected is the first defective item found?
17
Example (2)
At “busy time” a telephone exchange is very near capacity, so caller have difficulty placing their calls. It may be of interest to know the number of attempts necessary in order to gain a connection. Suppose that we let p=0.05 be the probability of a connection during busy time. We are interested in knowing the probability that 5 attempts are necessary for a successful call.
18
successes in n independent Bernoulli Trials.
• The geometric distribution is the distribution of the number of trials to get the first success in independent Bernoulli trials.
19
Poisson Distribution (1)
• Suppose we are counting the number of occurrences of an event in a given unit of time, distance, area, or volume.
• Example: • The number of car accidents in a day.
• The number of plants growing per acre.
• The number of defects per production line.
• The number of typing errors per page.
• The number of customers in a shop.
• Telephone calls received by the switchboard.
20
Example (1)
A sociologist observes from criminology data that bank robberies in one major city occur at the rate of 2.4 per day on the average. Calculate
a) the probability that exactly 3 robberies will occur in a day.
b) more than four robberies will occur in two- day period.
c) the expected number of robberies in four- day period.
22
Example (2)
The mean number of customers arrive at a bank is 4 customers in 10 minutes. Calculate the probability that:
a) exactly 2 customers will arrive at the bank counter in 10 minutes.
b) at least 1 customer will arrive in 10 minutes.
c) exactly 6 customers will arrive in 20 minutes
23
Example (3)
In the manufacturing of computer diskette, the number of defectives is 2 diskettes in 1000 diskettes. Assuming that the number of defective diskettes has a Poisson distribution, what is the probability of having
a) exactly 1 defective diskette in a purchase of 1000 diskettes?
b) at least 2 defective diskettes in a purchase of 2000 diskettes?
24
Example (4)
Suppose the number of tornadoes observed in a particular country during 1 year period has a Poisson distribution with mean 4. How many tornadoes do you expect in that country during
a) 5 years period?
b) 2 years period?
Normal Distribution
The normal distribution is the most used statistical distribution. The principal reasons are:
1. Normality arises naturally in many physical, biological, and social measurement situations.
2. Normality is important in statistical inference.
• The normal distribution is characterized by two parameters:
• the mean μ
• the standard deviation sigma.
• The mean is a measure of location or center and the standard deviation is a measure of scale or spread. The mean can be any value between ± infinity
• The standard deviation must be positive. Each possible value of μ and sigma define a specific normal distribution and collectively all possible normal distributions define the normal family.
• Any member of the normal family can be displayed by changing μ and sigma.
27
Characteristics of the Normal Distribution
1. It is bell shaped and is symmetrical about its mean.
2. It is asymptotic to the axis, i.e., it extends indefinitely in either direction from the mean.
3. It is a continuous distribution.
4. It is a family of curves, i.e., every unique pair of mean and standard deviation defines a different normal distribution. Thus, the normal distribution is completely described by two parameters: mean and standard deviation. See the following figure.
5. Total area under the curve sums to 1, i.e., the area of the distribution on each side of the mean is 0.5.
6. It is unimodal, i.e., values mound up only in the center of the curve.
7. 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.
28
Probability Density Function
The probability X is a specific value, i.e., P(X = x), is 0 since no area is above a singe point. It
follows that P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b).
29
The 68-95-99.7% Rule
• All normal density curves satisfy the following property which is often referred to as the Empirical Rule.
• 68% of the observations fall within 1 standard deviation of the mean, that is, between − and +
• 95% of the observations fall within 2 standard deviations of the mean, that is, between −2 and +2
• 99.7% of the observations fall within 3 standard deviations of the mean, that is, between −3and +3
• Thus, for a normal distribution, almost all values lie within 3 standard deviations of the mean.
• Remember that the rule applies to all normal distributions. Also remember that it applies only to normal distributions.
30
Standard Normal Distribution
• The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
• Normal distributions can be transformed to standard normal distributions by the formula:
= −
a) P( Z > 1.35)
32
Example (2)
• Graduate Management Aptitude Test (GMAT) scores are widely used by graduate schools of business as an entrance requirement. Suppose that in one particular year, the mean score for the GMAT was 476, with a standard deviation of 107. Assuming that the GMAT scores are normally distributed,
a) What is the probability that a randomly selected score from this GMAT falls between 476 and 650?
b) What is the probability of receiving a score greater than 750 on this GMAT test?
c) What is the probability of receiving a score of 540 or less on this GMAT test?
d) What is the probability of receiving a score between 440 and 330 on this GMAT test?
33
Example (3)
• Suppose that a tire factory wants to set a mileage guarantee on its new model called LA 50 tire. Life tests indicated that the mean mileage is 47,900, and standard deviation of the normally distributed distribution of mileage is 2,050 miles. The factory wants to set the guaranteed mileage so that no more than 5% of the tires will have to be replaced. What guaranteed mileage should the factory announce?
34
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Senior Lecturer Department of Statistics,
Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA,
Perak Branch, Tapah Campus.
Probability Distribution?
• A statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.
• This range will be bounded between the minimum and maximum possible values, but precisely where the possible value is likely to be plotted on the probability distribution depends on a number of factors.
• These factors include the distribution's mean (average), standard deviation, skewness, and kurtosis.
Discrete Distributio
Continuous Distribution
A statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.
Bernoulli Distribution (1)
• An experiment often consists of repeated trials, each with two possible outcomes that may be labelled as “success” or “failure”.
• The Bernoulli distribution is a discrete distribution of the outcome of a single trial with only two outcomes (0 or 1), with a probability of success and probability of failure (1- ).
• Example: • Coin Tosses: record how many coins land heads up and how many land tails
up.
• Births: How many boys are born and how many girls are born each day.
• Rolling Dice: The probability of a roll of two die resulting in a double six.
5
• A single trial
• The trial can result in one of two possible outcomes, (i.e: success or failure).
• P(success) = P or theta
• P(Failure) = 1-P or 1-theta.
Quick Test!
• In a Bernoulli trial, the probability of success is 0.3. What is the probability of failure?
• The probability of success in a Bernoulli trial is 0.3. What is the variance?
8
Example (1)
• Approximately 1 in 200 Malaysian adults are lawyers. One Malaysian adult is randomly selected. What is the distribution of the number of lawyers?
• = = ( 1
Binomial Distribution
• Data often arise in the form of counts or proportions which are realizations of a discrete random variable.
• A common situation is to record how many times an event occurs in n repetitions of an experiment, i.e., for each repetition the event either occurs (a "success") or it does not (a "failure").
10
1. There are n trials.
2. Each trial results in a success or a failure.
3. The probability of a success, p, is constant from trial to trial.
4. The trials are independent.
• An experiment satisfying these four conditions is called a binomial experiment.
• The outcome of this type of experiment is the number of successes, i.e., a count.
• The discrete variable X representing the number of successes is called a binomial random variable.
• The possible counts, X = 0, 1, 2, ..., n, and their associated probabilities define the binomial distribution, denoted by X~B(n,p).
11
Example (1)
Assume that 25% of fuses are defective, and the fuses in packages of six fuses are independently selected.
a) What is the probability that (exactly) two fuses in a package of six are defective?
b) What is the probability that fewer than two are defective?
c) Find the expected number of defective fuses.
13
Example (2)
A quality control inspection system requires that from each batch of items a sample of 10 is selected and tested. If 2 or more of the sample are defective the whole batch is rejected. If the probability of an item being defective is 0.05.
a) what is the probability of 2 defectives in the sample?
b) what is the probability of the batch being rejected?
14
Example (3)
In a certain country the percentage of illiterate adults is 30%. A random sample of 10 adults is selected at random in that country. Find the probability that
a) exactly five adults are illiterate
b) at least 4 adults are illiterate
Calculate the mean and variance for the illiterate adults in that country.
15
Example (1)
In a certain manufacturing process, it is known that, on the average, 1 in every 100 items is defective. What is the probability that the fifth item inspected is the first defective item found?
17
Example (2)
At “busy time” a telephone exchange is very near capacity, so caller have difficulty placing their calls. It may be of interest to know the number of attempts necessary in order to gain a connection. Suppose that we let p=0.05 be the probability of a connection during busy time. We are interested in knowing the probability that 5 attempts are necessary for a successful call.
18
successes in n independent Bernoulli Trials.
• The geometric distribution is the distribution of the number of trials to get the first success in independent Bernoulli trials.
19
Poisson Distribution (1)
• Suppose we are counting the number of occurrences of an event in a given unit of time, distance, area, or volume.
• Example: • The number of car accidents in a day.
• The number of plants growing per acre.
• The number of defects per production line.
• The number of typing errors per page.
• The number of customers in a shop.
• Telephone calls received by the switchboard.
20
Example (1)
A sociologist observes from criminology data that bank robberies in one major city occur at the rate of 2.4 per day on the average. Calculate
a) the probability that exactly 3 robberies will occur in a day.
b) more than four robberies will occur in two- day period.
c) the expected number of robberies in four- day period.
22
Example (2)
The mean number of customers arrive at a bank is 4 customers in 10 minutes. Calculate the probability that:
a) exactly 2 customers will arrive at the bank counter in 10 minutes.
b) at least 1 customer will arrive in 10 minutes.
c) exactly 6 customers will arrive in 20 minutes
23
Example (3)
In the manufacturing of computer diskette, the number of defectives is 2 diskettes in 1000 diskettes. Assuming that the number of defective diskettes has a Poisson distribution, what is the probability of having
a) exactly 1 defective diskette in a purchase of 1000 diskettes?
b) at least 2 defective diskettes in a purchase of 2000 diskettes?
24
Example (4)
Suppose the number of tornadoes observed in a particular country during 1 year period has a Poisson distribution with mean 4. How many tornadoes do you expect in that country during
a) 5 years period?
b) 2 years period?
Normal Distribution
The normal distribution is the most used statistical distribution. The principal reasons are:
1. Normality arises naturally in many physical, biological, and social measurement situations.
2. Normality is important in statistical inference.
• The normal distribution is characterized by two parameters:
• the mean μ
• the standard deviation sigma.
• The mean is a measure of location or center and the standard deviation is a measure of scale or spread. The mean can be any value between ± infinity
• The standard deviation must be positive. Each possible value of μ and sigma define a specific normal distribution and collectively all possible normal distributions define the normal family.
• Any member of the normal family can be displayed by changing μ and sigma.
27
Characteristics of the Normal Distribution
1. It is bell shaped and is symmetrical about its mean.
2. It is asymptotic to the axis, i.e., it extends indefinitely in either direction from the mean.
3. It is a continuous distribution.
4. It is a family of curves, i.e., every unique pair of mean and standard deviation defines a different normal distribution. Thus, the normal distribution is completely described by two parameters: mean and standard deviation. See the following figure.
5. Total area under the curve sums to 1, i.e., the area of the distribution on each side of the mean is 0.5.
6. It is unimodal, i.e., values mound up only in the center of the curve.
7. 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.
28
Probability Density Function
The probability X is a specific value, i.e., P(X = x), is 0 since no area is above a singe point. It
follows that P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b).
29
The 68-95-99.7% Rule
• All normal density curves satisfy the following property which is often referred to as the Empirical Rule.
• 68% of the observations fall within 1 standard deviation of the mean, that is, between − and +
• 95% of the observations fall within 2 standard deviations of the mean, that is, between −2 and +2
• 99.7% of the observations fall within 3 standard deviations of the mean, that is, between −3and +3
• Thus, for a normal distribution, almost all values lie within 3 standard deviations of the mean.
• Remember that the rule applies to all normal distributions. Also remember that it applies only to normal distributions.
30
Standard Normal Distribution
• The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
• Normal distributions can be transformed to standard normal distributions by the formula:
= −
a) P( Z > 1.35)
32
Example (2)
• Graduate Management Aptitude Test (GMAT) scores are widely used by graduate schools of business as an entrance requirement. Suppose that in one particular year, the mean score for the GMAT was 476, with a standard deviation of 107. Assuming that the GMAT scores are normally distributed,
a) What is the probability that a randomly selected score from this GMAT falls between 476 and 650?
b) What is the probability of receiving a score greater than 750 on this GMAT test?
c) What is the probability of receiving a score of 540 or less on this GMAT test?
d) What is the probability of receiving a score between 440 and 330 on this GMAT test?
33
Example (3)
• Suppose that a tire factory wants to set a mileage guarantee on its new model called LA 50 tire. Life tests indicated that the mean mileage is 47,900, and standard deviation of the normally distributed distribution of mileage is 2,050 miles. The factory wants to set the guaranteed mileage so that no more than 5% of the tires will have to be replaced. What guaranteed mileage should the factory announce?
34
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