4 MODULE IV-probability2 - MODULE IV Statistical Distributions Random Variables Statistical Distributions Using Statistics Expected Values of Discrete

4 MODULE IV-probability2 - MODULE IV Statistical...

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MODULE - IV Statistical Distributions Statistical Distributions
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Using Statistics Expected Values of Discrete Random Variables Sum and Linear Composite of Random Variables Bernoulli Random Variable The Binomial Random Variable The Poisson Distribution Continuous Random Variables Normal Distribution Random Variables & Random Variables & Statistical Distributions Statistical Distributions
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After studying this module you should be able to: After studying this module you should be able to: Distinguish between discrete and continuous random variables Explain how a random variable is characterized by its probability distribution Compute statistics about a random variable Compute statistics about a function of a random variable Compute statistics about the sum or a linear composite of a random variable Identify which type of distribution a given random variable is most likely to follow Solve problems involving standard distributions using formulas LEARNING OBJECTIVES LEARNING OBJECTIVES
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Consider the different possible orderings of boy (B) and girl (G) in four sequential births. There are 2*2*2*2=2 4 = 16 possibilities, so the sample space is: BBBB BGBB GBBB GGBB BBBG BGBG GBBG GGBG BBGB BGGB GBGB GGGB BBGG BGGG GBGG GGGG If girl and boy are each equally likely [P(G) = P(B) = 1/2], and the gender of each child is independent of that of the previous child, then the probability of each of these 16 possibilities is: (1/2)(1/2)(1/2)(1/2) = 1/16. Using Statistics
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Now count the number of girls in each set of four sequential births: BBBB (0) BGBB (1) GBBB (1) GGBB (2) BBBG (1) BGBG (2) GBBG (2) GGBG (3) BBGB (1) BGGB (2) GBGB (2) GGGB (3) BBGG (2) BGGG (3) GBGG (3) GGGG (4) Notice that: each possible outcome is assigned a single numeric value, all outcomes are assigned a numeric value, and the value assigned varies over the outcomes . The count of the number of girls is a random variable : A random variable, X, is a function that assigns a single, but variable, value to each element of a sample space. Random Variables
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Random Variables (Continued) BBBB BGBB GBBB BBBG BBGB GGBB GBBG BGBG BGGB GBGB BBGG BGGG GBGG GGGB GGBG GGGG 0 1 2 3 4 X X Sample Space Points on the Real Line
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Therefore, by a random variable we mean a real number X connected with the outcome of a random experiment E. E.g. if E consists of 2 tosses of a coin, we may consider the random variable which is the number of heads (0, 1 or 2). Outcome HH HT TH TT Value of X 2 1 1 0 RANDOM VARIABLE AND PROBABILITY DISTRIBUTION
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A random variable is an uncertain quantity whose value depends on chance. A random variable has a probability law – a rule that assigns probabilities to the different values of the random variable. The probability law, the probability assignment is called probability distribution of the random variable.
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