The word mean is sometimes used instead of expected

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The word mean is sometimes used instead of expected value, but one must be careful to distinguish the mean of a random variable from the sample mean of a sample of random variables. The variance of any random variable X is defined as the expected value of the squared deviation of X from E ( X ). That is, Var( X ) = E (( X - E ( X )) 2 ) . 2
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The square root of the variance of X is sometimes referred to as the standard deviation of X . If we expand the square in the above equation, and simplify using the linearity of expected values, we obtain another useful formula for the variance of X : Var( X ) = E ( X 2 - 2 XE ( X ) + E ( X ) 2 ) = E ( X 2 ) - E (2 XE ( X )) + E ( X ) 2 = E ( X 2 ) - 2 E ( X ) E ( X ) + E ( X ) 2 = E ( X 2 ) - E ( X ) 2 . Using the definition of the expected value, the above equality leads us to a simple expression for the variance of X in terms of the pmf or pdf of X . If X is a discrete random variable with pmf f X , we have Var( X ) = E ( X 2 ) - E ( X ) 2 = X x x 2 f X ( x ) - X x xf X ( x ) ! 2 , while if X is a continuous random variable with pdf f X , we have Var( X ) = E ( X 2 ) - E ( X ) 2 = Z -∞ x 2 f X ( x )d x - Z -∞ xf X ( x )d x 2 . Note that to obtain both of these expressions, we have used the definition of the expectation of g ( X ), where g ( x ) = x 2 . 3 The uniform distribution The uniform distribution provides one example of how a continuous random vari- able might behave. A random variable X is said to be uniformly distributed be- tween a and b , written X U ( a, b ), if it is continuous with pdf f X given by f X ( x ) = 0 for x < a 1 b - a for a x b 0 for x > b. It is tempting to think of a U ( a, b ) random variable as taking any value between a and b with equal probability. While technically true, this intuition is misleading. A U ( a, b ) random variable is continuous, and therefore takes any fixed value with probability zero – just like any other continuous random variable. A more accurate statement is to say that a random variable X has the U ( a, b ) distribution if (1) for any interval ( c, d ) contained in ( a, b ), the probability that X lies in ( c, d ) is exactly 3
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the ratio of the length of ( c, d ) to the length of ( b, a ), and (2) for any interval ( c, d ) that lies completely outside of ( a, b ), the probability that X lies in ( c, d ) is zero. It is obvious from the definition of the U ( a, b ) pdf f X that R -∞ f X ( x )d x = 1. This is because the area under f X is a rectangle of width ( b - a ) and height 1 / ( b - a ). We can also calculate the expected value and variance of a U (0 , 1) random variable using the formulas given in the previous section. For the expected value, we have E ( X ) = Z -∞ xf X ( x )d x = Z b a x b - a d x = 1 2 b 2 - 1 2 a 2 b - a = 1 2 ( a + b ) , which is just the midpoint of the interval ( a, b ). To obtain the variance, it is helpful to first calculate E ( X 2 ): E ( X 2 ) = Z -∞ x 2 f X ( x )d x = Z b a x 2 b - a d x = 1 3 b 3 - 1 3 a 3 b - a = 1 3 ( a 2 + ab + b 2 ) . Combining our expressions for E ( X 2 ) and E ( X ) yields Var( X ) = E ( X 2 ) - E ( X ) 2 = 1 3 ( a 2 + ab + b 2 ) - 1 2 ( a + b ) 2 = 1 12 ( b - a ) 2 .
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