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Theorems about mean, variance - PowerPoint PPT Presentation

Theorems about mean, variance. Properties of mean, variance for one random variable X, where a and b are constant: • E[aX+b] = aE[X] + b • Var(aX+b) = a 2 Var(X) • Var(X) = E[X 2 ] – (E[X]) 2

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• Properties of mean, variance for one random variable X, where a and b are constant: • E[aX+b] = aE[X] + b • Var(aX+b) = a2Var(X) • Var(X) = E[X2] – (E[X])2

• Theorem. Let X and Y be independent random variables and let g and h be real valued functions of a single real variable.

• Theorem. For random variables X1, X2, ... , Xn, defined on the same sample space, and for constants a1, a2, ... , an, we have

• Consider an exponential r. v. with λ = 1. The density is:

• Note that the mean µ and the median m are different. The density has a lot of weight “in the tail” which causes the mean to be larger. We say that this density is “skewed to the right”.

m = 0.693

µ=1

• Suppose we are given a random variable X with some unknown probability distribution. We want to estimate the basic parameters of this distribution, like the expectation of X and the variance of X.

• The usual way to do this is to observe n independent variables all with the same distribution as X. To estimate the unknown mean  of X, we use the sample mean described on the next slide. The value of the observations yield a value for the sample mean which is used as an estimate for . In a similar way, the sample variance (discussed later) is used to estimate the variance of X.

• Let X1,X2,…,Xn be independent and identically distributed random variables having c. d. f. F and expected value μ. Such a sequence of random variables is said to constitute a sample from the distribution F. The sample mean is denoted by and is defined by

• By using the theorem on the previous slide, we have

• Thus, the expected value of the sample mean is μ, the mean of the distribution. For this reason, is said to be an unbiased estimator of μ.

• The random variable is an example of a statistic. That is, it is a function of the observations which does not depend on the unknown parameter μ.

• Recall that a Bernoulli random variable Xi is defined by

• Since Xi is a discrete random variable, we have

• Let X be a binomial random variable with parameters (n, p). Then X = X1+ X2+…+ Xn where each Xi is Bernoulli. By the theorem from the previous slide, which agrees with the direct computation we did earlier.

Definition. The covariance between r.v.’s X and Y, denoted by Cov(X,Y), is defined by

• Theorem.

• Corollary. If X and Y are independent, then Cov(X, Y) = 0.

• Example. Two dependent r. v.'s X and Y might have Cov(X, Y) = 0. Let X be uniform over (–1, 1) and let Y = X2.

• Let X and Y be random variables. Then

• If we take Yj = Xj, then (iv) implies that

• If Xi and Xj are independent when i and j differ, then the latter equation becomes

• Let X1,X2,…,Xn be independent and identically distributed random variables having c. d. f. F, expected value μ, and variance 2. Let be the sample mean. The random variable is called the sample variance.

• Using the results from previous slides, we have

• Recall that a Bernoulli random variable Xi is defined by Also, Var(Xi) = p – p2 as an easy computation shows (taking advantage of the fact that

• Let X be a binomial random variable with parameters (n, p). Then X = X1+ X2+…+ Xn where each Xi is Bernoulli. By the result from a previous slide,

• Upon combining the above results, we have which agrees with our earlier result.

• For random variables X and Y, Cov(X,Y) might be positive, negative, or zero.

• If Cov(X, Y) > 0, then X and Y decrease together or increase together. In this case, we say X and Y are positively correlated.

• If Cov(X, Y) < 0, then X increase while Y decreases or vice versa. In this case, we say X and Y are negatively correlated.

• If Cov(X, Y) = 0, we say that X and Y are uncorrelated. Recall that uncorrelated random variables may be dependent, however.