Probability distributions. 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
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AS91586 Apply probability distributions in solving problems
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.
To estimate mean and standard deviation, students need to know that:
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).
Estimating the mean:
Estimating the standard deviation:
Words remembered in Kim’s Game problem, demonstrating understanding of the way a probability distribution changes as the parameter values change.
Mean = 13.1
SD = 2.4
Mean = 9.0
SD = 2.8
Text messages sent in a day by stage one university students problem, demonstrating understanding of the way a probability distribution changes as the parameter values change.
Mean = 38 messages
SD = 57 messages
Mean = 10.4 pairs
SD = 8.9 pairs
Mean = students5.9 words
SD = 2.5 words
Mean = 7.0 words
SD = 23 words
How do you introduce:
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.
Uniform: roll of one die
Triangular: Sum of two dice
What are the units on the vertical axis for a continuous probability function?
The vertical axis measures the rate probability/x, which is called probability density.
Probability density is only meaningful in terms of area.
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?
Uniform models are used for modellingdistributions when the only information you have are maximum and minimum.
Triangular models are used for modellingdistributions when the only information you have are maximum, minimum and average (could be the mode).
Stephen J Gould