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Probability distributions Probability distributions

Probability distributions - PowerPoint Presentation

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Probability distributions - PPT Presentation

AS91586 Apply probability distributions in solving problems NZC level 8 Investigate situations that involve elements of chance calculating and interpreting expected values and standard deviations of discrete random variables ID: 269567

distributions distribution standard probability distribution distributions probability standard deviation estimate model normal binomial students time bus discrete probabilities poisson

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Slide1

Probability distributions

AS91586 Apply probability distributions in solving problemsSlide2

NZC level 8

Investigate situations that involve elements of chance

calculating and interpreting expected values and standard deviations of discrete random variables

applying distributions such as the Poisson, binomial, and normalSlide3

AS91586

Apply probability distributions in solving problems

Methods include a selection from those related to:

discrete and continuous probability

distributions

mean

and standard deviation of random variables

distribution of true probabilities versus distribution of model estimates of probabilities versus distribution of experimental estimates of

probab

ilities

.Slide4

AO8-4 TKI

Calculating and interpreting expected values and standard deviations of discrete random variables:

A statistical data set may contain discrete numerical variables. These have

frequency distributions that can be converted to empirical probability distributions. Distributions from both sources have the same set of possible features (centre, spread, clusters, shape, tails, and so on) and we can calculate the same measures (mean, SD, and so on) for them.

Makes a reasonable estimate of mean and standard deviation from a plot of the distribution of a

discrete random variable

.

Solves and interprets solutions of problems involving calculation of mean, variance and standard deviation from a

discrete probability distribution

.

Solves and interprets solutions of problems involving linear transformations and sums (and differences) of

discrete random variables

.Slide5

Applying distributions such as the Poisson, binomial, and normal:

They

learn that some situations that satisfy certain conditions can be modelled mathematically. The model may be

Poisson

, binomial

,

normal

,

uniform, triangular, or others, or be derived from the situation being investigated.

Recognises situations in which probability distributions such as

Poisson

,

binomial

, and

normal

are appropriate models, demonstrating understanding of the assumptions that underlie the distributions.

Selects and uses an appropriate distribution to model a situation in order to solve a problem involving probability.

Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the

link between probabilities and areas under density functions for continuous outcomes (for example,

normal

, triangular, or uniform, but nothing requiring integration

).Slide6

Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the way a probability distribution changes as the parameter values change.

Selects and uses an appropriate distribution to solve a problem involving finding and using estimates of parameters.

Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of

the relationship between true probability (unknown and unique to the situation), model estimates (theoretical probability) and experimental estimates.

Uses a distribution to estimate and calculate probabilities, including by

simulation

.Slide7

AS 3.14 summary

Includes expected value and standard deviation (and variance).

Includes sums and differences (and linear combinations) of random variables.

Includes binomial, Poisson and normal, but also includes uniform, triangular distributions and experimental distributions.

Requires consideration of context as well as appearance of the distribution when selecting a model.Slide8

Looking at distributions

(simulated normal distribution)

Small samples do not always have distributions like the population they come from.

When looking at distributions, a sample of 30 is much too small to give a good picture of the whole population distribution.Slide9

Looking at distributions

(simulated normal distribution)

Large samples do have distributions like the population they come from.

When looking at distributions, a sample of about 200 is sufficient to give a picture of the whole population distribution.Slide10

Estimating mean and standard deviation

To estimate mean and standard deviation, students need to know that:

The mean is pulled towards extreme values

The SD is stretched by extreme values

If the distribution is approximately normal, the mean is the middle, and the SD is roughly 1/6th the range (97.8% within μ ± 3σ).Slide11

Estimating mean and standard deviation for any distribution

Estimating the mean:

Estimate the median and adjust towards extreme values.

Estimating the standard deviation:

Estimate the median distance from the mean and adjust it (stretch it if there are extreme values). Slide12

Mean =

12.3 years

SD =

1.8 years

Estimate the mean and standard deviation of the

age of students completing the census@school survey.Slide13

Words remembered in Kim’s Game

Mean = 13.1

SD = 2.4

Mean = 9.0

SD = 2.8Slide14

Mean =

38 messages

SD =

57 messages

Text messages sent in a day by stage one university studentsSlide15

Mean =

10.4 pairs

SD =

8.9 pairs

Number of pairs of shoes owned by stage one university studentsSlide16

Mean =

5.9 words

SD =

2.5 words

Mean =

7.0

words

SD =

23

wordsSlide17

Introducing distributions

How do you introduce:

Binomial

Poisson

NormalUniform

Triangular distributions?Slide18

Introducing the binomial distribution

Combinations and permutations are still in the curriculum, so you can still teach them if you want to.

You can teach the binomial distribution without using combinations by using trees. Introduce the binomial distribution as a shortcut for complicated trees.Slide19

Chuck-a-luck

A gambling game played at carnivals, played against a banker.

A

player pays a dollar to play and rolls 3 dice.

If no 4s are rolled, the player loses. Otherwise the player gets back one dollar for every 4 rolled and gets their original dollar back.Slide20
Slide21

Once students see the pattern emerge, they can start to

generalise

it, using Pascal’s triangle or an understanding of combinations to get the coefficients.

For some students, it may be enough to know that the calculator is a shortcut method for working out probabilities from trees like these.

Introducing the binomial distributionSlide22

Poisson distributions

Hokey Pokey ice-cream – is Tip Top really the best?

Choc chip cookies: number of choc chips visible on an area of cookie (

Farmbake

Triple Choc works

well - do white chips and dark chips separately).Slide23

Discrete uniform and triangular distributions

Uniform: roll of one die

Triangular: Sum of two diceSlide24

Continuous probability graphs

What are the units on the vertical axis for a continuous probability function? Slide25

Continuous probability graphs are

probability density functions

The vertical axis measures the rate probability/x, which is called probability density.

Probability density is only meaningful in terms of area.Slide26

bus waiting time (1)

The downtown inner link bus in Auckland arrives at a stop every ten minutes, but has no set times.

If I turn up at the bus

stop,

how long will I expect to wait for a bus?

What will the distribution of wait times look like?Slide27

a

b cSlide28

0.1

0 10Slide29

Which is more likely: a wait of between 2 and 5 minutes, or a wait of more than 6 minutes, measured to the nearest minute?

0.1

0 10Slide30

Bus waiting time (2)

My own bus route (277) runs only every half

hour, and isn’t as reliable as the inner link.

I know that the bus is most likely to appear on time, but could in fact turn up at any time between the time it is due and half an hour later.Slide31

What is the best model for wait time, given the available information?Slide32

In the real world:

Uniform models are used

for

modelling

distributions when the only information you have are maximum and minimum.Triangular models are used for

modelling

distributions when the

only information you have are maximum, minimum and average (could be the mode).Slide33

a

b cSlide34

What is the probability that I will have to wait longer than 20 minutes for a bus?

1

15

0 30Slide35

My interpretation of AS 3.14

Expect more questions giving experimental data to be fitted to a theoretical model.

Expect more evaluation of how well a theoretical model fits experimental data.

Expect more interpretation of the application of a model in context.Slide36

Teaching and learning

Students should:

record their hunch every time you start an investigation, and compare the results to their hunch.

always consider the context and the distribution you would expect in that context, as well as their observations of the data available.

Estimate the mean and the standard deviation every time they look at a distribution (write down the estimate, then check to see how close they were).Slide37

Learning could start with:

Questions to investigate, and gathering data: What is the probability that at least 4 people in

a

class have the same

birth month? Data in tables: which distribution (if any) would you use to model it? Estimate probabilitiesData in graphs: estimate mean and standard deviation, which distribution (if any) would model it? Estimate probabilities.Slide38

A learning activity

From Teaching Statistics: a bag of tricks (

Gelman

and Nolan)Slide39
Slide40
Slide41
Slide42
Slide43

What do you notice?

Students tend to group their guessed histogram into large groups.

Different bin widths will give different estimates of probability.

What else do

you notice?Slide44

Misunderstanding of probability may be the greatest of all impediments to scientific literacy.

Stephen J Gould