lect13

lect13 - CONTINUOUS RANDOM VARIABLES UNIFORM RANDOM...

This preview shows pages 1–5. Sign up to view the full content.

CONTINUOUS RANDOM VARIABLES Outline CONTINUOUS RANDOM VARIABLES UNIFORM RANDOM VARIABLES EXPECTATION THE NORMAL DISTRIBUTION 1 / 16 Xinghua Zheng Lect 13: Continuous random variables; uniform and normal

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CONTINUOUS RANDOM VARIABLES CONTINUOUS RANDOM VARIABLES A random variable X is said to have a continuous distribution if there exists a nonnegative function f such that P ( X B ) = R B f ( x ) dx for any set B of real numbers. In particular, R -∞ f ( x ) dx = . Interpretation of f : if f is continuous at a point x , then lim Δ 0 P ( X [ x , x + Δ]) Δ = lim Δ 0 R x x f ( y ) dy Δ = . Shorthand for this: P ( x X x + dx ) = f ( x ) dx . f is called the probability density function (for short, pdf or just density) of X . The function F : ( -∞ , ) [ 0 , 1 ] deﬁned by F ( x ) = P ( X x ) is called the cumulative distribution function (cdf for short) of X . 2 / 16 Xinghua Zheng Lect 13: Continuous random variables; uniform and normal
CONTINUOUS RANDOM VARIABLES RELATIONSHIP BETWEEN DENSITY AND CDF If X has density f , then you get its cdf by integration : F ( x ) = P ( X ( -∞ , x ]) = Z x -∞ f ( y ) dy . Conversely, if the cdf F of X is differentiable, or if F is everywhere continuous and differentiable except at ﬁnitely many points, then by the fundamental theorem of calculus , P ( X ( x , z ]) = F ( z ) - F ( x ) = Z x z F 0 ( y ) dy for all -∞ < x < z < , so X has density f = . 3 / 16 Xinghua Zheng Lect 13: Continuous random variables; uniform and normal

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CONTINUOUS RANDOM VARIABLES UNIFORM RANDOM VARIABLES A random variable X is said to be uniformly distributed over the interval [ a , b ] , written X Uniform ( a , b ) , if it has a density f X which is constant on that interval, and equal to 0 outside that interval: c = What is the graph of the cdf F X of X ? For a α < β b , P ( α < X β ) = . Put U = ( X - a ) / ( b - a ) . For 0 u 1, the cdf of U is F U ( u ) = P ± X - a b - a u ² = P ( a X a + u ( b - a )) = , so U has density f U ( u ) = F 0 U ( u ) = ( , if 0 u 1 ; , otherwise .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 16

lect13 - CONTINUOUS RANDOM VARIABLES UNIFORM RANDOM...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online