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Published byBuddy Fox Modified over 9 years ago
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Statistical Experiment A statistical experiment or observation is any process by which an measurements are obtained
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Examples of Statistical Experiments Counting the number of books in the College Library Counting the number of mistakes on a page of text Measuring the amount of rainfall in your state during the month of June
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Random Variable a quantitative variable that assumes a value determined by chance
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Discrete Random Variable A discrete random variable is a quantitative random variable that can take on only a finite number of values or a countable number of values. Example: the number of books in the College Library
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Continuous Random Variable A continuous random variable is a quantitative random variable that can take on any of the countless number of values in a line interval. Example: the amount of rainfall in your state during the month of June
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Probability Distribution an assignment of probabilities to the specific values of the random variable or to a range of values of the random variable
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Probability Distribution of a Discrete Random Variable A probability is assigned to each value of the random variable. The sum of these probabilities must be 1.
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Probability distribution for the rolling of an ordinary die xP(x) 1 2 3 4 5 6
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Features of a Probability Distribution xP(x) 1 2 3 4 5 6 Probabilities must be between zero and one (inclusive) P(x) =1
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Probability Histogram P(x) 1 2 3 4 5 6 ||||||||||||||
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Mean and standard deviation of a discrete probability distribution Mean = = expectation or expected value, the long-run average Formula : = x P(x)
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Standard Deviation
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Finding the mean: xP(x) x P(x) 0.3 1.3 2.2 3.1 4.1 0.3.4.3.4 1.4 = x P(x) = 1.4
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Finding the standard deviation xP(x) x – ( x – ) 2 ( x – ) 2 P(x) 0.3 1.3 2.2 3.1 4.1 – 1.4 – 0.4.6 1.6 2.6 1.96 0.16 0.36 2.56 6.76.588.048.072.256.676 1.64
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Standard Deviation 1.28
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Linear Functions of a Random Variable If a and b are any constants and x is a random variable, then the new random variable L = a + bx is called a linear function of a random variable.
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If x is a random variable with mean and standard deviation , and L = a + bx then: Mean of L = L = a + b Variance of L = L 2 = b 2 2 Standard deviation of L = L = the square root of b 2 2 = b
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If x is a random variable with mean = 12 and standard deviation = 3 and L = 2 + 5x Find the mean of L. Find the variance of L. Find the standard deviation of L. L = 2 + 5 Variance of L = b 2 2 = 25(9) = 225 Standard deviation of L = square root of 225 =
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Independent Random Variables Two random variables x 1 and x 2 are independent if any event involving x 1 by itself is independent of any event involving x 2 by itself.
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If x 1 and x 2 are a random variables with means and and variances and If W = ax 1 + bx 2 then: Mean of W = W = a + b Variance of W = W 2 = a 2 1 2 + b 2 2 Standard deviation of W = W = the square root of a 2 1 2 + b 2 2
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Given x 1, a random variable with 1 = 12 and 1 = 3 and x 2 is a random variable with 2 = 8 and 2 = 2 and W = 2x 1 + 5x 2. Find the mean of W. Find the variance of W. Find the standard deviation of W. Mean of W = 2(12) + 5(8) = 64 Variance of W = 4(9) + 25(4) = 136 Standard deviation of W= square root of 136 11.66
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