The Probability Distribution for a random variable can beB) Continuous Random VariablesUniformSome interesting facts about the NORMAL DISTRIBUTION:e.g. IQ of 110 (Equivalent Z-scoreTopic (8) – POPULATION DISTRIBUTIONS 8-1 Topic (8) – POPULATION DISTRIBUTIONS So far: We’ve seen some ways to summarize a set of data, including numerical summaries. We’ve heard a little about how to sample a population effectively in order to get good estimates of the population quantities of interest (e.g. taking a good sample and calculating the sample mean as a way of estimating the true but unknown population mean value) We’ve talked about the ideas of probability and independence. Now we need to start putting all this together in order to do Statistical Inference, the methods of analyzing data and interpreting the results of those analyses with respect to the population(s) of interest. The Probability Distribution for a random variable can be a table or a graph or an equation.Topic (8) – POPULATION DISTRIBUTIONS 8-2 Let’s start by reviewing the ideas of frequency distributions for populations using categorical variables. QUALITATIVE (NON-NUMERIC) VARIABLES For a random variable that takes on values of categories, the Probability distribution is a table showing the likelihood of each value. EXAMPLE Tree species found in a boreal forest. For each possible species there would a probability associated with it. E.g. suppose there are 4 species and three are very rare and one is very common. A probability table might look like: Species Probability1 0.01 2 0.03 3 0.08 4 0.88 All 1.00 We interpret these values as the probability that a random selection would result in observing that species. We could also draw a bar chart but it would be fairly non-informative in this instance since one value is so much larger than the others! An equation cannot be developed since the values that the variable takes on are non-numeric.Topic (8) – POPULATION DISTRIBUTIONS 8-3 QUANTITATIVE (NUMERICAL) VARIABLES A) Discrete Random Variables Recall that a discrete random variable is one that takes on values only from a set of isolated (specific) numbers. The relative frequency distribution for a discrete random variable (also sometimes called a probability mass function) is a list of probabilities for each possible value that the variable can take on. BERNOULLI DISTRIBUTION Suppose the scientist studying the tree species overlaid a grid of square quadrats over the region of interest and then recorded whether any tree was in the quadrat or not. Hence, the random variable is binary, i.e. only two outcomes presence (1) or absence (0). The Bernoulli distribution describes the probability of each outcome: Pr(X=1) = π Pr(X=0) = 1 – π The mean for a Bernoulli variable is π and the variance is π(1-π).Topic (8) – POPULATION DISTRIBUTIONS 8-4 POISSON DISTRIBUTION Suppose the scientist studying the tree species overlaid a grid of square quadrats over the region of interest and then counted the number of hickory trees in each quadrat. The histogram of the number of trees per quadrat for all of the quadrats might look like Tree Count0.100.200.300.400 1 2 3 4 5 6 7 8 9 10 11Quantilesmaximum quartilemedianquartile minimum100.0%99.5%97.5%90.0%75.0%50.0%25.0%10.0%2.5%0.5%0.0% 11.000 8.000 7.000 5.000 4.000 3.000 2.000 1.000 0.000 0.000 0.000MomentsMeanStd DevStd Error MeanUpper 95% MeanLower 95% MeanNSum Weights 2.999 1.750 0.025 3.048 2.951 5000.000 5000.000Topic (8) – POPULATION DISTRIBUTIONS 8-5 Since we have sampled the entire population (the set of counts for every quadrat in the region), this histogram represents the probability distribution of the random variable X = ”number of trees/quadrat”. In general, the Poisson distribution is a common probability distribution for counts per unit time or unit area or unit volume. The graph can also be described using an equation known as the Poisson Distribution Probability Mass Function. It gives the probability of observing a specific count (x) in any randomly selected quadrat as !)Pr(xexXxµµ−== where )1)(2)(3)...(2)(1(!−−=xxxx and ,...2,1,0=x. In order for this distribution to be a valid probability distribution, we require that the total probability for all possible values equal 1 and that every possible value have a probability associated with it. ∑∑=−====,...2,1,0,...2,1,01!)Pr(XxXxexXµµTopic (8) – POPULATION DISTRIBUTIONS 8-6 and 0!)Pr( ≥==−xexXxµµ The mean of the Poisson distribution is µ and the variance is µ as well. DISCRETE UNIFORM DISTRIBUTION: every discrete value that the random variable can take on has the same probability of occurring. For example, suppose a researcher is interested in whether the number of setae on the first antennae of an insect is random or not. Further, the researcher believes that there must be at least 1 seta and at most 8. Then s/he is postulating that every value between 1 and 8 are equally likely to be observed in a random draw of an insect from the population (or equivalently, that there are equal numbers of insects with 1, 2, …, or 8 setae in the population). Such a distribution is known as the Discrete Uniform Distribution. Let K be the total number of distinct values that the random variable can take on (e.g. the set {1, 2, …, 8} contains K = 8 distinct values). Then, KxX1)Pr( == for x = 1, 2, …, 8Topic (8) – POPULATION DISTRIBUTIONS 8-7 In addition, the mean for this particular discrete uniform is 5.4836===∑Kxµ and the variance is 25.5)5.4(22=−=∑Kxσ. Also, it is easy to see that the probabilities sum to 1 as required. Finally, the graph of the distribution looks like a rectangle: 0 2 4 6 8Topic (8) – POPULATION DISTRIBUTIONS 8-8 B) Continuous Random Variables Recall that a continuous random variable is one that can take on any value from an interval on the number line. Now, for relative frequency distributions: Fact 1: They show the frequencies of the values of the variable of interest in a set of data: TIME95.090.085.080.075.070.065.060.055.050.045.040.050403020100Std. Dev = 12.80 Mean = 71.0N = 222.00 where the data have been assigned to specific groupings (bins or categories). The height of each bar is proportional to the relative frequency in the data set
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