Stats for Engineers Lecture 5
Summary From Last Time
Binomial Distribution ๐ (๐=๐ )=ยฟ(๐๐)๐๐ (1โ๐ )๐โ๐
๐=๐๐Mean and variance ๐ 2=๐๐(1โ๐ )
Probability of number of success when you do Bernoulli trials
Poisson distribution
Probablily of randomly occurring events, given average number is
๐ (๐=๐ )=๐โ๐๐๐
๐!
Mean and variance
Is approximation to Binomial when n is large and p is small
Discrete Random Variables
Continuous Random Variables๐ (๐โค ๐โค๐ )=โซ
๐
๐
๐ (๐ฅ โฒ )๐๐ฅ โฒProbability Density Function (PDF)
Uniform distribution1
2 1
2
0
otherwise1 x
๐ (๐ฅ )=ยฟ
1 2 3 4
18%23%
5%
54%
Poisson or not?
Which of the following is most likely to be well modelled by a Poisson distribution?
1. Number of trains arriving at Falmer every hour
2. Number of lottery winners each year that live in Brighton
3. Number of days between solar eclipses
4. Number of days until a component fails
Are they Poisson? Answers:
1. Number of trains arriving at Falmer every hour
NO, (supposed to) arrive regularly on a timetable not at random
2. Number of lottery winners each year that live in Brighton
Yes, is number of random events in fixed interval
3. Number of days between solar eclipses
NO, solar eclipses are not random events and this is a time between random events, not the number in some fixed interval
4. Number of days until a component failsNO, random events, but this is time until a random event, not the number of random events
If a Poisson process has constant average rate , the mean after a time is .
What is the probability distribution for the time to the first event?
Exponential distribution
Poisson - Discrete distribution: P(number of events)
Exponential - Continuous distribution: P(time till first event)
Time between random events / time till first random event ?
Exponential distributionThe continuous random variable has the Exponential distribution, with constant rate parameter if:
Occurrence 1) Time until the failure of a part. 2) Separation between randomly happening events
- Assuming the probability of the events is constant in time:
๐ (๐ฆ ) ๐=1
๐ฆ
๐ (๐ฆ )={๐๐โ๐ ๐ฆ , ๐ฆ>00 ,โง๐ฆ<0
Relation to Poisson distribution
The probability of no-occurrences in time is
If is the pdf for the first occurrence, then the probability of no occurrences is
ยฟ1โ๐ (first occurrence has happened by ๐ก)ยฟ1โโซ
0
๐ก
๐ (๐ก )๐๐ก
โ1โโซ0
๐ก
๐ (๐ก )๐๐ก=๐โ๐๐ก โโซ0
๐ก
๐ (๐ก )๐๐ก=1โ๐โ๐๐ก
Solve by differentiating both sides respect to assuming constant ,
โ ๐ (๐ก )=๐๐โ๐๐กThe time until the first occurrence (and between subsequent occurrences) has the Exponential distribution, parameter .
If a Poisson process has constant average rate , the mean after a time is .
๐ (no occurrence by ๐ก)
Example
On average lightening kills three people each year in the UK, So the rate is .
Assuming strikes occur randomly at any time during the year so is constant, time from today until the next fatality has pdf (using in years)
๐ (๐ก)
๐ก
E.g. Probability the time till the next death is less than one year?
โซ0
1
๐ (๐ก )๐๐ก=โซ0
1
3๐โ3 ๐ก ๐๐ก
ยฟ [3๐โ 3๐ก
โ3 ]0
1
ยฟโ๐โ 3+1โ0.95
1 2
53%
48%
Exponential distribution
A certain type of component can be purchased new or used. 50% of all new components last more than five years, but only 30% of used components last more than five years. Is it possible that the lifetimes of new components are exponentially distributed?
Question from Derek Bruff
1. YES2. NO
Exponential distribution
A certain type of component can be purchased new or used. 50% of all new components last more than five years, but only 30% of used components last more than five years. Is it possible that the lifetimes of new components are exponentially distributed?
Exponential distribution models time between independent randomly occurring events, where frequency of events is independent of time.
i.e. probability of failing in the first 5 years has to be same as the probability of failing in any other period of 5 years. No memory property.
The observed lifetimes imply that instead the failure rate must increase with time
NOT exponential
Mean and variance of exponential distribution
๐=13
๐=3
๐ 2=โซโโ
โ
๐ฆ2 ๐ (๐ฆ )๐๐ฆโ๐2=โซ0
โ
๐ฆ2๐๐โ๐ ๐ฆ ๐๐ฆโ 1๐2 =[โ ๐ฆ2๐โ๐ ๐ฆ ]0
โ+2โซ
0
โ
๐ฆ ๐โ๐ ๐ฆ ๐๐ฆโ 1๐2 =0+2
๐๐โ
1๐2 =
1๐2
๐๐
Example: Reliability
The time till failure of an electronic component has an Exponential distribution and it is known that 10% of components have failed by 1000 hours.
(a) What is the probability that a component is still working after 5000 hours?
(b) Find the mean and standard deviation of the time till failure.
Answer Let Y = time till failure in hours;
๐ (๐ โค1000 )=โซ0
1000
๐๐โ๐ ๐ฆ(a) First we need to find
ยฟ [โ๐โ๐ ๐ฆ ]01000
ยฟ1โ๐โ1000 ๐
๐ (๐ โค1000 )=0.1โ1โ๐โ1000๐=0.1โ๐โ 1000๐=0.9โโ1000๐=ln 0.9=โ0.10536โ๐โ1.05ร10โ 4
If is the time till failure, the question asks for :
๐ (๐>5000 )=โซ5000
โ
๐๐โ๐ ๐ฆ๐๐ฆ
ยฟ [โ๐โ๐ ๐ฆ ]5000
โ
ยฟ๐โ5000 ๐โ 0.59
(b) Find the mean and standard deviation of the time till failure.
Mean = = 9491 hours.Answer:
Standard deviation == = 9491 hours
1 2 3 4
17%
27%27%
39%
Is it exponential?
Which of the following random variables is best modelled by an exponentialdistribution?
Question adapted from Derek Bruff
1. The distance between defects in an optical fibre
2. The number of days between someone winning the National Lottery
3. The number of fuses that blow in the UK today
4. The hours of sunshine in Brighton this week assuming an average of 7.2hrs/day
Is it exponential?
Which of the following random variables is best modelled by an exponentialdistribution?
1. The distance between defects in an optical fibre
- YES: continuous distribution that is the separation between independent random events (the location of the defects)
2. The number of days between someone winning the National Lottery
- NO: continuous (if you allow fractional days), but draws happen regularly on a schedule
3. The number of fuses that blow in the UK today
- NO: this is a discrete distribution โ the number of events is a Poisson distribution (exponential is the distribution of times between events)
4. The hours of sunshine in Brighton this week assuming an average of 7.2hrs/day
- NO: This is a continuous variable, but not the time between independent random events
Normal distribution
The continuous random variable has the Normal distribution if the pdf is:
mean standard deviationNote: The distribution is also sometimes called a Gaussian distribution
X lies between - 1.96 and + 1.96 with probability 0.95
i.e. X lies within 2 standard deviations of the mean approximately 95% of the time.
๐
โซโโ
โ
๐ (๐ฅ )๐๐ฅ=1
[see notes for proof]
Occurrence of the Normal distribution 1) Quite a few variables, e.g. distributions of sizes, measurement errors, detector noise. (Bell-shaped histogram). 2) Sample means and totals - see later, Central Limit Theorem. 3) Approximation to several other distributions - see later.
If has a Normal distribution with mean and variance , write