notes7 - MTH4106 Introduction to Statistics Notes 7 Spring...

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Unformatted text preview: MTH4106 Introduction to Statistics Notes 7 Spring 2012 Continuous random variables If X is a random variable (abbreviated to r.v.) then its cumulative distribution function (abbreviated to c.d.f.) F is defined by F ( x ) = P ( X 6 x ) for x in R . We write F X ( x ) if we need to emphasize the random variable X . (Take care to distinguish between X and x in your writing.) We say that X is a continuous random variable if its cdf F is a continuous function. In this case, F is differentiable almost everywhere, and the probability density function (abbreviated to p.d.f.) f of F is defined by f ( x ) = d F d x for x in R . Again, we write f X ( x ) if we need to emphasize X . If a < b and X is continuous then P ( a < X 6 b ) = F ( b )- F ( a ) = Z b a f ( x ) d x = P ( a 6 X 6 b ) = P ( a < X < b ) = P ( a 6 X < b ) . Moreover, F ( x ) = Z x- ∞ f ( t ) d t and Z ∞- ∞ f ( t ) d t = 1 . The support of a continuous random variable X is { x ∈ R : f X ( x ) 6 = } . The median of X is the solution of F ( x ) = 1 2 ; the lower quartile of X is the solution of F ( x ) = 1 4 ; and the upper quartile of X is the solution of F ( x ) = 3 4 . The top decile 1 of X is the solution of F ( x ) = 9 / 10; this is the point that separates the top tenth of the distribution from the lower nine tenths. More generally, the n-th percentile of X is the solution of F ( x ) = n / 100. The expectation of X is defined by E ( X ) = Z ∞- ∞ x f ( x ) d x . Similarly, if g is any real function then E ( g ( X )) = Z ∞- ∞ g ( x ) f ( x ) d x . In particular, the n-th moment of X is E ( X n ) = Z ∞- ∞ x n f ( x ) d x and Var ( X ) = E ( X 2 )- μ 2 = Z ∞- ∞ ( x- μ ) 2 f ( x ) d x , where μ = E ( X ) . Two special continuous random variables Uniform random variable U ( a , b ) also known as uniform [ a , b ] Let a and b be real numbers with a < b . A uniform random variable on the interval [ a , b ] is, roughly speaking, “equally likely to be anywhere in the interval”. In other words, its probability density function is some constant c on the interval [ a , b ] (and zero outside the interval). What should the constant value c be? The integral of the p.d.f. is the area of a rectangle of height c and base b- a ; this must be 1, so c = 1 / ( b- a ) . Thus, the p.d.f. of the random variable X ∼ U ( a , b ) is given by f X ( x ) = if x < a , 1 ( b- a ) if a < x < b , if x > b . So the support of X is the interval [ a , b ] , as we would expect. By integration, we find that the c.d.f. is F X ( x ) = if x < a , ( x- a ) ( b- a ) if a 6 x 6 b , 1 if x > b . 2 (Strictly speaking, the support of X is the open interval ( a , b ) , because f X ( a ) and f X ( b ) are not defined, as F X is not differentiable at x = a or at x = b . However, the results from Calculus that we need to use in Theorem 7 are usually stated in terms of closed intervals. For the purposes of MTH4106 Introduction to Statistics...
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This note was uploaded on 03/12/2012 for the course MTH 4106 taught by Professor R.a.bailey during the Spring '12 term at Queen Mary, University of London.

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notes7 - MTH4106 Introduction to Statistics Notes 7 Spring...

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