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Probability theory The department of math of central south university Probability and Statistics Course group §3.2 Continuous random variable 1、The definitions for Continuous R.V 2、Properties for the density function of Continuous R.V 3、Properties for the distribution function of Continuous R.V 4、Common continuous R.V 1、The definitions for Continuous R.V Definiton 3.2 A distribution function ,F(x),which describes the probability distribution for a random variable ξ(ω) and satisfies that there exists a non-negative Integrable function p(x) to establish the equation F ( x) x P( y)dy then ξ(ω)is called a continuous random variable ,p(x) is called the probability density function of ξ(ω), F(x) is called the distribution function of ξ(ω). 2、Properties for the density function of Continuous R.V p( x) 0, x (1) (2) p( x)dx 1 For any function, p(x),is called a probability density function only if it satisfies the obove properties (1)、(2) Example: sin x p ( x) 0 x [0, / 2] others For p( x) 0 (1) (2) p( x)dx sin xdx cos x 2 0 /2 0 Here P(x) is certainly a probability density function 1 Geometric interpretation for distribution finction F(x) and density function p(x) p(x) F (x ) F(1) F(-1) -1 0 1 x 3、Properties for the distribution function of Continuous R.V According the definition of continuous random variable,the distribution function F(x) of R.V ξ(ω) has the following good properties (1) p (a b) F (b) F (a ) b a p ( x)dx p( x)dx b p ( x)dx a (2) If F(x) is abosolute continuous,then F(x) is continuous everywhere , has derivative in continuous point of p(x) and F ( x) p( x) (3) Especially,for any constant c, p( c) F (c) F (c 0) lim c h 0 c h =0 p( x)dx Example 1 ζ is acontinuous R.Vwith the following distribution function x0 0 F ( x ) Ax 2 0 x 1 1 1 x (1) What value does A gets? (2) What is the probability that ζ gets value in interval (0.3,0.4), that is , P(0.3< ζ <0.7)? (3) What is the density function of X? x0 0 F ( x ) Ax 2 0 x 1 1 1 x Solution (1) Beause F(x) is left continuous at point x=1, We get lim F ( x) F (1) x 1 That is Then lim Ax2 F (1) 1 x 1 A=1 x0 0 2 F ( x ) Ax 0 x 1 1 1 x (2) P(0.3< ζ <0.7)=F(0.7)-F(0.3) =0.72- 0.32=0.4 x0 0 2 F ( x ) Ax 0 x 1 1 1 x (3) p( x) F ( x) 2 x 0 0 x 1 Others 4、Common continuous random variables (1). Uniform distribution a and b are constants,and a<b,R.Vζis called a uniform distribution random variable whose density function is expressed as follows 1 a xb p( x) b a 0 Others Briefly , we write ~ U (a, b) For x a, it is obvious that F ( x) P( x) 0 For a x b F ( x) P ( x) x a For x b ,obviously 1 xa dx ba ba F ( x ) P ( x ) x a 1 dx ba b a x 1 dx 0dx 1 b ba Hence the distribution function of Uniform distributioncan be expressed as follows F ( x) P ( x) 0 x a b a 1 xa a xb xb The density function and distribution function of uniform distribution p ( x) a b x b x F( x) a (2). Normal distribution One of the most commonly observed random variables is one whose probability density curve is characterized by two paramters μand σ2,the density function is p( x) 1 e 2 ( x )2 2 2 x I t is called the normal random variable marked by ζ ~ N (μ,σ2 ) Clearly, p(x) satisfies the necessary two conditions (1) ( 2) 1 p( x) e 2 p( x ) y ( x )2 2 2 1 e 2 x 1 2 0 x e y2 2 ( x )2 2 2 dx dy 1 The probability density curve is symmetrical (left half is mirror image of the right half) and bell shaped,as shown in the following figure. σ μ Normal distribution is applicated widely . Generally speaking, if a target is affected by many factors the role played by each factor is not great, then the target subjects to normal distribution. example : When we Measure something , because of precision instruments, the visual and psychological factors, outside interference and other factors, the measurement results generally subject to normal distribution; measurement error also subject to normal distribution. In addition, the physical size of the adult such as height, weight, a class in a certain area of the diameter of trees, shells fall, certain products of a similar size, and so on are subject to normal distribution. On the other hand, the normal distribution has a good nature under certain conditions, many normal distribution can be used to approximate some other distributions, and some can be derived through the normal distribution, therefore, the normal distribution in the theoretical study is also important In fact, the normal distribution was first introduced in the early 19th century by Gaussian (Gauss) who was studying measurement error ,therefore normal distribution is also known as the error distribution or Gaussian distribution; Two parameters of p (x) : — Location parameter That is ,for given and fifferent , p( x) keep same shape and lacate in different places — Shape parameter For given and different ,p( x) locate in a position with diffetent shapes If 1< 2 , 1 2 1 1 2 2 For 1, the probability that random variable get value close to μ is larger than that of 2 Inflection point corresponding to x = μ 1 is closer to straight line x = μ than that x= μ 2 The probabity density curve of normal distribution 1.The curve is symmetrical to p(x) N (-5 , 1.2 ) 0.3 0.25 0.2 line x= μ ,that means p (μ + x) = p(μ - x) 0.15 0.1 0.05 x= -8 -7 -6 -5 -4 2. p (x) gets the maximal value when x=μ -3 3.Curve has infection points at points x= μ± The relation between density curve of normal distribution and When is small p(x) 0.3 When become large 0.25 0.2 0.15 0.1 0.05 x= -8 -7 -6 -5 -4 -3 Applications If the random variable ζ is affected by a number of independent random factors, and the effect of each individual is minor, and these effects can be superimposed, then ζ subjects to normal distribution. many examples can normally be described Various measurement error; People's physical characteristics; The size of plant products; Crop harvest; The height of ocean wave height ; The tensile strength of metal; and so on A important normal distribution:---------N (0,1) — standard normal distribution Suppose ζ ~ N (0,1),the distributon function of ζ should be ( x) 1 2 x e t2 2 dt x It can easily be seen that : ( x) 1 ( x) 0.4 0.3 0.2 0.1 -x P(| X | a) 2(a) 1 x 0.4 (0) 0.5 0.3 0.2 0.1 -3 -2 -1 1 2 3 For a general normal distribution :ζ ~ N ( , 2) 1 x F ( x) e 2 The distribution is : Let t s ( t ) 2 2 2 x F ( x ) P(a b) F (b) F (a) P ( a ) 1 F (a ) b a a 1 dt Example2 Assume that ζ ~ N(1,4) , what is the probability of ζ in [0 ,1.6] Solution 1.6 1 0 1 P(0 1.6) 2 2 0.3 0.5 0.3 [1 0.5] 0.6179 [1 0.6915] 0.3094 Example 3 It is known that ~ N (2, 2 ) , and P( 2 < ζ < 4 ) = 0.3,What is P ( ζ < 0 )? Solution 0 2 1 2 P( 0) 42 22 2 P(2 4) 0.3 (0) 2 0.8 P( 0) 0.2 Graphic method Solution The left shaded area shown in the following figure represents the probabity in question 0.2 0.3 0.15 0.1 0.2 0.05 -2 2 4 6 3 principle Example 4 It is known that ζ ~ N ( α , 2), what is the probability of | | 3 ? Solution P(| | 3 ) P( 3 3 ) 3 3 3 3 23 1 2 0.9987 1 0.9974 The αth percentile is a score that 100(1- α)% of the distribution lies below it Supposeζ ~ N (0,1) ,0 < < 1,if z suffice the express P( z ) Here z is called the upper percentile. 0.4 We must remenber several important upper percentile 0.3 0.2 0.1 z z0.05 1.645 z0.025 1.96 (3). Exponential distribution Suppose the random variable ζhas such a density function with parameter λthat e x x 0 p( x) 0 x0 ζis called an exponential distribution with parameterλ,which can be remembered as follows ~ ( ) ~ ( ) We can easily get the distribution function of ζ Firstly,When x is non-positive,it is obviously that F ( x) P( x) 0 When x is greater than zero x F ( x) P( x) e x dx 0 x de x (1 e x ) 0 Thus , exponential distribution function is 1 e x x 0 F ( x) x0 0 Exponential distribution which is similar to geometric distribution has "no memory." In fact, a random variable ζ that subjects to an exponential distribution with parameters λ means P( s t s ) P( s t , s ) e ( s t ) e s P( s ) e t Here, s and t are arbitrary and s> 0, t> 0, 休息片刻继续