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C1100__17

# C1100__17 - Random variables and probability distributions...

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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.
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 p +p +…+p =1
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
7 Notations for random variables We denote the random variable by uppercase letters (e.g., X,V,Z ) We denote the values that the random variable gets by lowercase letters (e.g., x i , v i , z i ). The probability that a random variable X get a certain value x is denoted by: p(X=x ) , or, in short , p(x ) . Value of X P(x i ) 0 0.25 1 0.5 2 0.25 P(X=0)=0.25 P(X=1)=0.5 P(X=2)=0.25 Remember that the sum of the probabilities must equal 1!!!

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8 Example A probability distribution is given in the accompanying table with the additional information that the even values of X are equally likely. Determine the missing entries of the table. Value of X P(x) 1 0.2 2 3 0.2 4 5 0.3 6 Answer: The sum of the probabilities in the table: 0.2+0.2+0.3=0.7 The remaining 0.3 probability is equally divided between the values 2,4,and 6 0.1 0.1 0.1
9 Example Here is the probability distribution p(x) for grade X of a randomly chosen student from a certain class. X =0 represents an F, X =4 represents an A.

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C1100__17 - Random variables and probability distributions...

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