2293 words - 10 pages

By the way, parametric distribution means normal, binomial distribution etc..

Probability Distribution

A listing of all the outcomes of an experiment and the probability associated with each outcome.

Mean of probability Distribution

μ= Σ[xP(x)

Variance of probability Distribution

σ2= Σ[x-μ2Px]

Number of Cars (x) | Probability (x) | (x-μ) | (x-μ)2 | x-μ2P(x) |

0 | 0.1 | 0 – 2.1 | 4.41 | 0.441 (4.41*0.1) |

1 | 0.2 | 1 – 2.1 | 1.21 | 0.242 |

2 | 0.3 | 2 – 2.1 | 0.01 | 0.003 |

3 | 0.3 | 3 – 2.1 | 0.81 | 0.243 |

4 | 0.3 | 4 – 2.1 | 3.61 | 0.361 |

| 1.0 (total) | | | σ2=1.290 |

To derive μ=00.10+10.20+30.30+40.10 = 2.1

Discrete Random Variable

A random ...view middle of the document...

g.

There are 5 flights. Suppose probability that any flight arrives late is 0.20.

a) What is the probability that none of the flights are late today?

b) What is the probability that exactly 1 flight is late today?

Binomial Distribution

Number of late flights | Probability | Workings |

0 | 0.3277 | 5C0 π0 (1-0.2)5-0 |

1 | 0.4096 | 5C1 π1 (1-0.2)5-1 |

2 | 0.2048 | |

3 | 0.0512 | |

4 | 0.0064 | |

5 | 0.0003 | |

Total | 1 | |

Ans:

a) P(0) = 0.3277

b) P(1) = 0.4096

Number of late flights(x) | Probability p(X) | xP(x) | x-μ | (x-μ)2 | x-μ2P(x) |

0 | 0.3277 | 0 | -1 | 1 | 0.3277 |

1 | 0.4096 | 0.4096 | 0 | 0 | 0 |

2 | 0.2048 | 0.4096 | 1 | 1 | 0.2048 |

3 | 0.0512 | 0.1536 | 2 | 4 | 0.2048 |

4 | 0.0064 | 0.0256 | 3 | 9 | 0.0576 |

5 | 0.0003 | 0.0015 | 4 | 16 | 0.0048 |

Total | 1 | μ=1 | | | σ2=0,7997 |

Instead of doing all these lame table and calculations:

Use the formula would suffice,

μ=nπ

μ=50.2=1.0

σ2=nπ(1-π)

σ2=50.21-0.2=0.8

Poisson probability distribution

* The random variable is the number of times some event occurs during a defined interval.

* The probability of the event is proportional to the size of the interval

* The intervals which do not overlap and are independent. Intervals can be time, distance, area or volume.

Poisson can use to describe distributions in errors in data entry, scratches on car, defective parts on shipments.

Px= μxe-ux!

X is the number of occurrences

Mean & Variance of Poisson

μ=nπ

Mean and variance are the same.

Difference Poisson & Binomial (MUST KNOW)

For binomial there is a fixed number of trial. There can be only a fixed number of success (x)

For Poisson, the x can be infinite values. For e.g. number of bags lost in a flight….can be infinite. ‘

* If a mean or average probability of an event happening per unit time/per page/per mile cycled etc., is given, and you are asked to calculate a probability of n events happening in a given time/number of pages/number of miles cycled, then the Poisson Distribution is used.

E.g. It is ESTIMATED that the probability is……. The word ‘estimate means that there’s no exact probability.

* If, on the other hand, an exact probability of an event happening is given, or implied, in the question, and you are asked to calculate the probability of this event happening k times out of n, then the Binomial Distribution must be used.

Answers

1. A typist makes on average 2 mistakes per page. What is the probability of a particular page having no errors on it?

We have an average rate here: lambda = 2 errors per page.

We don't have an exact probability (e.g. something like "there is a

probability of 1/2 that a page contains errors").

Hence, Poisson distribution.

(lambda t) = (2 errors per page * 1 page) = 2.

Hence P0 = 2^0/0! * exp(-2) = 0.135.

2. A computer crashes once...

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