web1000__16 - Random variables and probability...

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1 Random variables and probability distributions
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2 Random variables are a key concept for statistical inference We know that in order to use results from a sample to infer about a larger population, we must avoid bias and chose a sample at random. If our sample is random, then the sample mean, the sample proportion, and other statistics that we calculate from the sample are called: random variables . The behavior of such variables is an essential step on the way to performing statistical inference.
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3 Random variables A random variable X associates a numerical value with each elementary outcome of an experiment Example Let X be the number of boys born in a family of 2 children. List the possible values of X X =1 means that there is one boy in the family 2 children in the family X – number of boys BB 2 BG 1 GB 1 GG 0
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4 Probability distribution of discrete variables In order to understand the pattern of behavior of a random variable, we need to know its possible values and how likely they are to occur. The probability distribution of a discrete random variable X is a list of the distinct numerical values of X , along with their associated probabilities. X x 1 x 2 x 3 …… x k Probability p 1 p 2 p 3 …… p k 0≤p i ≤1 i=1,…,k
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5 Example If X represents the number of boys in a family of 2 children, find the probability distribution of X. The probability of BB is: p(first is a boy and second is a boy) = p(first is a boy)× p(second is a boy)=0.5×0.5=0.25 P(BG)= P(GB)= P(GG)= Family with 2 children X – number of boys Probability BB 2 BG 1 GB 1 GG 0 Since the 2 events are independent 0.25 0.25 0.25 0.25 BG 1 0.25 GB 1 0.25 GG 0 0.25
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6 Example - continued Now we can specify the probability of each value of X Family with 2 children X – number of boys Probability BB 2 0.25 BG 1 0.25 GB 1 0.25 GG 0 0.25 Value of X Probability 0 0.25 1 0.25+0.25=0.5 2 0.25
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This note was uploaded on 11/03/2009 for the course IST IST taught by Professor N/a during the Spring '09 term at Anadolu University.

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web1000__16 - Random variables and probability...

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