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© Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T3 Scotland Statistics Mean and Variance of Discrete Random Variables ©Teachers Teaching with Technology (Scotland) Higher Statistics Mean and Variance of Discrete Random Variables Let X be a discrete random variable taking values x1 , x2 ,....., xn with probabilities p1 , p2 ,......, pn . The variance of X, denoted by σ 2 , is the number σ 2 = ( x1 − µ ) 2 p1 + ( x2 − µ ) 2 p2 + ........... + ( xn − µ ) 2 pn The standard deviation σ of x is the square root of the variance When given the probability distribution of a random variable X, the Ti-83 list facility can be used to find the mean and standard deviation. Example 1 Let the output of the random variable X denote the number of defective computer parts in a shipment of 400. The following table gives the probability density function (pdf) of X X Pdf(x) 0 0.02 1 0.2 2 0.3 3 0.3 4 0.1 Compute the mean and standard deviation. Solution: Input the values given in L1 and L2 (Fig 1) Fig 1 Calculate the mean and standard deviation Use STAT CALC 1: 1-Var Stats (Fig 2) followed by the list name (Fig 3 ). Fig 2 Fig 3 This gives 3 ©T Scotland Higher Statistics (Mean and Variance) Page 1 of 2 5 0.08 Exercises 1. A random variable X has a probability density function given by: X Pdf(x) -2 0.3 -1 0.2 0 0.1 1 0.3 2 0.1 Compute the mean and standard deviation 2. Bluetits always lay three eggs. The number of eggs, X, which hatch has the following probability distribution. X Pdf(x) 0 0.1 1 0.3 2 0.4 3 0.2 Compute the mean and standard deviation 3 ©T Scotland Higher Statistics Page 2 of 2