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4.3 RANDOM VARIABLES Discrete and continuous random variables Normal distributions as probability distributions 4.4 MEANS AND VARIANCES OF RVs Rules for means of random variables Rules for variances of random variables The law of large numbers Discrete random variables A discrete random variable can take on only a certain number of possible values. • Discrete random variables The probability distribution for a discrete random variable can be given by a list of the possible values, and the probability of each one. value of X probability 2 3 4 5 6 7 8 9 10 11 12 .028 .056 .083 .111 .139 .167 .139 .111 .083 .056 .028 Discrete random variables “How many heads will I get if I toss two coins?” sample space = {TT, TH, HT, HH} 0.5 Number of heads 0 1 2 0.4 Probability .25 .50 .25 0.3 0.2 0.1 0 0 1 2 Continuous random variables Continuous random variables can take all possible values in a particular interval. The probability distribution of a continuous random variable is described by a density curve. The probability that the random variable falls within a certain interval is given by the area under the curve in that interval. Normal distributions as probability distributions “What is the probability that a woman selected at random is 67 inches tall or taller?” The heights of adult American women are approximately normally distributed with m=64.5 inches and s=2.5 inches. z = (67-64.5) / 2.5 = +1.00 Table A tells us that about 16% of a normal curve lies to the right of z = +1.00, so the probability is about .16. 4.4 MEANS and VARIANCES of RANDOM VARIABLES Means of random variables The mean of a discrete random variable is equal to the sum of each possible value multiplied by its probability. mX x1p1 x 2p 2 x kpk For example, the mean of the number of heads we see when we toss two coins is mX (0).25 (1).50 (2).25 1 Means of random variables The mean of a continuous random variable is the point at which the density curve would balance. The mean of any random variable is also known as its expected value. Statistical estimation The law of large numbers says that if you want an extremely accurate estimate of m, all you have to do is to draw an extremely large random sample, and your sample mean will be acceptably close to m. In other words, “the average results of many independent observations are stable and predictable.” Rules for means 1 The expected value of a linear transformation is the linear transformation of the expected value: m(a bX) a bmX 2 The expected value of a sum is the sum of the expected values: m(X Y) mX mY The variance of a discrete random variable The variance of a discrete random variable is the sum of each possible squared deviation multiplied by its probability: s X (x1 - mX ) p1 (x k - mX ) p k 2 2 2 For example, the variance of the number of heads we see when we toss two coins is ó 2 X (0 - 1) .25 (1 1) .50 (2 - 1) .25 0.5 2 ó X 0.707 2 2 Rules for variances 1 The variance of a linear transformation is the original variance times the square of the multiplier: ó 2 (a bX) b ó 2 2 X 2 The variance of a sum is the sum of the variances (if X and Y are independent): ó 2 (X Y) ó 2 X ó 2 Y The variance of a difference is the sum of the variances (if X and Y are independent): ó 2 (X Y) ó 2 X ó 2 Y