复旦大学：《概率论》精品课程教学资源（习题答案）统计学中的三大分布 SOME SPECIFIC PROBABILITY DISTRIBUTIONS

SOME SPECIFIC PROBABILITY DISTRIBUTIONS 1. NORMAL RANDOM VARIABLES with mean u and variance a2(abbreviated by x a N[u, 02] if the density function of x is given bb 1. 1. Probability Density Function. The random variable X is said to be normally distribute f(x;μ,a2) The normal probability density function is bell-shaped and symmetric. The figure below shows the probability distribution function for the normal distribution with a u=0 and o=l. The areas between the two lines is 0.68269. This represents the probability that an observation lies within one standard deviation of the mear FIGURE 1. Normal Probability density Function μ=0

SOME SPECIFIC PROBABILITY DISTRIBUTIONS 1. Normal random variables 1.1. Probability Density Function. The random variable X is said to be normally distributed with mean µ and variance σ2 (abbreviated by x ∼ N[µ, σ2] if the density function of x is given by f (x ; µ, σ2) = 1 √ 2πσ2 · e −1 2 ( x−µ σ ) 2 (1) The normal probability density function is bell-shaped and symmetric. The figure below shows the probability distribution function for the normal distribution with a µ = 0 and σ =1. The areas between the two lines is 0.68269. This represents the probability that an observation lies within one standard deviation of the mean. Figure 1. Normal Probability Density Function -1 1 .1 .2 .3 Μ = 0, Σ = 1 Date: August 9, 2004. 1

SOME SPECIFIC PROBABILITY DISTRIBUTIONS The next figure below shows the portion of the distribution between -4 and 0 when the mean is d o is equal to tw FIGURE 2. Normal Probability Density Function Showing P(-4<I<O Probability Between Limits is 0. 30233 0 0.18 0.16 0.14 0.04 4 1. 2. Properties of the normal random variable. (x)=u, var(x)=g b: The density is continuous and symmetric about u c: The population mean, median, and mode coinci d: The range is unbound e: There are points of inflection atμ±σ f: It is completely specified by the two parameters u and a g: The sum of two independently distributed normal random variables is normally distributed If Y= aX1 BX2 +y where X1 NN(1, 01) and X2 NN(a2, 022) and X1 and X2 are 1.3. Distribution function of a normal random variable f(s;u, o2)d Here is the probability density function and the cumulative distribution of the normal distribution with u=0 and o= 1

2 SOME SPECIFIC PROBABILITY DISTRIBUTIONS The next figure below shows the portion of the distribution between -4 and 0 when the mean is one and σ is equal to two. Figure 2. Normal Probability Density Function Showing P(−4 <x< 0) −8 −6 −4 −2 0 2 4 6 8 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Probability Between Limits is 0.30233 Density Critical Value 1.2. Properties of the normal random variable. a: E(x) = µ, Var(x) = σ2. b: The density is continuous and symmetric about µ. c: The population mean, median, and mode coincide. d: The range is unbounded. e: There are points of inflection at µ ± σ. f: It is completely specified by the two parameters µ and σ2. g: The sum of two independently distributed normal random variables is normally distributed. If Y = αX1 + βX2 + γ where X1 ∼ N(µ1,σ1 2) and X2 ∼ N(µ2,σ2 2) and X1 and X2 are independent, then Y ∼ N(αµ1 + βµ2 + γ; α2σ2 1 + β2σ2 2). 1.3. Distribution function of a normal random variable. F(x ; µ, σ2) = P r (X ≤ x) = Z x −∞ f (s ; µ, σ2 )ds (2) Here is the probability density function and the cumulative distribution of the normal distribution with µ = 0 and σ = 1

SOME SPECIFIC PROBABILITY DISTRIBUTIONS IGURE 3. Normal pdf and cdf Probability Density Function Cumulative distribution Function 08 1.4. Evaluating probability statements with a normal random variable. If x NN(u, 02) N(0,1) E(

SOME SPECIFIC PROBABILITY DISTRIBUTIONS 3 Figure 3. Normal pdf and cdf −10 −5 0 5 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Probability Density Function X f(X) −10 −5 0 5 10 0 0.2 0.4 0.6 0.8 1 Cumulative Distribution Function X F(X) 1.4. Evaluating probability statements with a normal random variable. If x ∼ N(µ,σ2) then, Z = X−µ σ ∼ N(0, 1) E (Z) = E  X−µ σ  = 1 σ · (E(X) − µ)=0 V ar (Z) = V ar  X−µ σ  = 1 σ2 V ar(X − σ) = σ2 σ2 = 1 (3)

SOME SPECIFIC PROBABILITY DISTRIBUTIONS Pr(a≤x≤b)=Pr(a-≤x-≤b-p) 0.1)-F(=0. area below 4. Probability of Intervals 3 2 b 1.96 a=1.6 We can then merely look in tables for the distribution function of a N(0, 1) variable 1.5. Moment generating function of a normal random variable. The moment generating function for the central moments is as follows Mx(t) The first central moment is E(Xx-p)=#(=)h=0 The second central moment is

4 SOME SPECIFIC PROBABILITY DISTRIBUTIONS Consequently, P r(a ≤ x ≤ b) = P r (a − µ ≤ x − µ ≤ b − µ) = P r h a − µ σ ≤ x − µ σ ≤ b − µ σ i = F b − µ σ ; 0, 1  − F ￾ a − µ σ ; 0, 1
= area below (4) Figure 4. Probability of Intervals b - Μ €€€€€€€€€€€€€€€€€ Σ a - Μ €€€€€€€€€€€€€€€€ Σ .1 .2 .3 Μ = 0, Σ = 1 b = -1.96 a = 1.6 We can then merely look in tables for the distribution function of a N(0,1) variable. 1.5. Moment generating function of a normal random variable. The moment generating function for the central moments is as follows MX (t) = e t2 σ2 2 . (5) The first central moment is E (X − µ ) = d dt  e t2 σ2 2  |t = 0 = t σ2  e t2 σ2 2  |t = 0 = 0 (6) The second central moment is

SOME SPECIFIC PROBABILITY DISTRIBUTIONS E(x-)2=(=+)h=0 (#a(=)+a2(f)k= The third central moment is E(x-)2=需( (t2 () (2°(=)+2to +3 g6 +3tσ 0 The fourth central moment i E(X-p)4=(-)h σ8(e)+3t2o +326(e (=) +6t2a +3a(e2 2. CHI-SQUARE RANDOM VARIABLE 2.1. Probability Density Function. The random variable X is said to be a chi-square random variable with v degrees of freedom (abbreviated x(u)] if the density function of X is given by )=2r( 20 (10) 0 otherwise here r(.is the gamma function defined by r(r)=fo ur 0 (11) Note that for positive integer values of r, r(r)=(r-1)!

SOME SPECIFIC PROBABILITY DISTRIBUTIONS 5 E (X − µ )2 = d2 dt2  e t2 σ2 2  |t = 0 = d dt  t σ2  e t2 σ2 2  |t= 0 =  t2 σ4  e t2 σ2 2  + σ2  e t2 σ2 2  |t= 0 = σ2 (7) The third central moment is E (X − µ )3 = d3 dt3  e t2 σ2 2  |t= 0 = d dt  t2 σ4  e t2 σ2 2  + σ2  e t2 σ2 2  |t = 0 =  t 3 σ6  e t2 σ2 2  + 2 t σ4  e t2 σ2 2  + t σ4  e t2 σ2 2  |t= 0 =  t 3 σ6  e t2 σ2 2  + 3 t σ4  e t2 σ2 2  |t = 0 = 0 (8) The fourth central moment is E (X − µ )4 = d4 dt4  e t2 σ2 2  |t = 0 = d dt  t 3 σ6  e t2 σ2 2  + 3 t σ4  e t2 σ2 2  |t = 0 =  t 4 σ8  e t2 σ2 2  + 3 t 2 σ6  e t2 σ2 2  + 3 t 2 σ6  e t2 σ2 2  + 3 σ4  e t2 σ2 2  |t = 0 =  t4 σ8  e t2 σ2 2  + 6 t2 σ6  e t2 σ2 2  + 3 σ4  e t2 σ2 2  |t = 0 = 3 σ4 (9) 2. Chi-square random variable 2.1. Probability Density Function. The random variable X is said to be a chi-square random variable with ν degrees of freedom [abbreviated χ2(ν) ] if the density function of X is given by f (x ; ν) = 1 2 ν 2 Γ ( v 2 ) x ν−2 2 e −x 2 0 0 (11) Note that for positive integer values of r, Γ(r) = (r - 1)!

SOME SPECIFIC PROBABILITY DISTRIBUTIONS The following diagram shows the pdf and cdf for the chi-square distribution with parameters v FIGuRE 5. Ch df and cdf Probability Density Function Cumulative Distribution Function 0 0 0.02 X 2.2. Properties of the chi-square random variable 2.2.1. x and N(0, 1). Consider n independent random variables IfX;~N(0,1) (12) It can also be shown that IfX;~N(0,1) then >(Xi-X)N x(n-1) 13) because this is the sum of (n-1)independent random variables given that X and(n-1)of the x's 222.x2andN(1,a2) IfX1~N(μ,a2) 1.2 th (14) and x-12 ~x2(n-1) 2.2.3. Sums of chi-square random variables. If yi and y2 are independently distributed as x(v1) (1+D2)

6 SOME SPECIFIC PROBABILITY DISTRIBUTIONS The following diagram shows the pdf and cdf for the chi-square distribution with parameters ν =10. Figure 5. Chi-square pdf and cdf 0 10 20 30 0 0.02 0.04 0.06 0.08 0.1 Probability Density Function X f(X) 0 10 20 30 0 0.2 0.4 0.6 0.8 1 Cumulative Distribution Function X F(X) 2.2. Properties of the chi-square random variable. 2.2.1. χ2 and N(0,1). Consider n independent random variables. If Xi ∼ N (0, 1) i = 1, 2, ... , n then Pn i=1 X2 i ∼ χ2(n) (12) It can also be shown that If Xi ∼ N (0, 1) i = 1, 2, ... , n then Pn i=1 (Xi − X¯) 2 ∼ χ2(n − 1) (13) because this is the sum of (n-1) independent random variables given that X¯ and (n-1) of the x’s are independent. 2.2.2. χ2 and N(µ,σ2). If Xi ∼ N (µ, σ2) i = 1, 2, ... , n then Xn i=1 Xi − µ σ 2 ∼ χ2(n) (14) and Xn i=1 Xi − X¯ σ 2 ∼ χ2 (n − 1) 2.2.3. Sums of chi-square random variables. If y1 and y2 are independently distributed as χ2(ν1) and χ2(ν2), respectively, then y1 + y2 ∼ χ2(ν1 + ν2). (15)

SOME SPECIFIC PROBABILITY DISTRIBUTIONS 2.2.4. Momen hi-square random variables Mean(x(u))=v= degrees of freedom (x2(u) (16) 2.3. The distribution function of x(v) s tabulated in most statistics and econometrics texts 2.4. Moment generating function. The moment generating function is as follows (1-2t) The first moment is E(x)=是(m=m)h t=0 (19) 3. THE STUDENTS T RANDOM VARIABLE This distribution was published by William Gosset in 1908. His employer, Guinness Breweries equired him to publish under a pseudonym, so he chose" Student 3. 1. Relationship of Students t-Distribution to Normal Distribution. The ratio t=xt0.1) has the Student's t density function with v degrees of freedom where the standard normal variate the numerator is distributed independently of the x variate in the denominator. Tabulations of the associated distribution function are included in most statistics and econometrics books note nat it is sy 3.2. Probability Density Function. The density of Student's t distribution is given b t2 √Dr(兰)

SOME SPECIFIC PROBABILITY DISTRIBUTIONS 7 2.2.4. Moments of chi-square random variables. M ean (χ2 (ν)) = ν = degrees of freedom V ar (χ2 (ν)) = 2 ν Mode (χ2 (ν)) = ν − 2 (16) 2.3. The distribution function of χ2(ν). F(x; ν) = Z x 0 f (s; ν)ds (17) is tabulated in most statistics and econometrics texts. 2.4. Moment generating function. The moment generating function is as follows MX(t) = 1 (1 − 2 t) υ/2 ,t < 1 2 (18) The first moment is E ( X ) = d dt  1 ( 1 − 2 t ) υ/2  |t = 0 =  υ ( 1 − 2 t ) ( υ + 1)/2  |t = 0 = υ (19) 3. The Student’s t random variable This distribution was published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under a pseudonym, so he chose ”Student.” 3.1. Relationship of Student’s t-Distribution to Normal Distribution. The ratio t = N(0, 1) qχ2(ν) ν (20) has the Student’s t density function with ν degrees of freedom where the standard normal variate in the numerator is distributed independently of the χ2 variate in the denominator. Tabulations of the associated distribution function are included in most statistics and econometrics books. Note that it is symmetric about origin. 3.2. Probability Density Function. The density of Student’s t distribution is given by: f (t; ν ) = Γ ￾ ν + 1 2
√πν Γ ￾ ν 2
 1 + t 2 ν  −( ν +1) 2 − ∞ <t< ∞ (21)

SOME SPECIFIC PROBABILITY DISTRIBUTIONS The following diagram shows the pdf and cdf for the Student's t-distribution with parameter v FIGURE 6. Student's t distribution pdf and cdf Probability Density Function Cumulative Distribution Function 0.15 0.05 The following diagram shows the cdf for the Student's t-distribution with parameters v=10 and FIGURE 7. Student's t-distribution with alternative parameter levels f(x) =3.3 0.2 0.1 2

8 SOME SPECIFIC PROBABILITY DISTRIBUTIONS The following diagram shows the pdf and cdf for the Student’s t-distribution with parameter ν = 10. Figure 6. Student’s t distribution pdf and cdf −10 −5 0 5 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Probability Density Function X f(X) −10 −5 0 5 10 0 0.2 0.4 0.6 0.8 1 Cumulative Distribution Function X F(X) The following diagram shows the cdf for the Student’s t-distribution with parameters ν = 10 and ν = 3. Figure 7. Student’s t-distribution with alternative parameter levels -4 -2 2 4 0.1 0.2 0.3 fHxL v = 3 v = 10

SOME SPECIFIC PROBABILITY DISTRIBUTIONS 3.3. Moments of student's t-distribution (t()=0 (22) 4 THE F (F ISHER VARIANCE RATIO) STATISTIC 4.1. Distribution Function. If x21(v1)and x22(v2) are independently distributed chi-square vari- ates. then X(2 x2(2) has the F density with v1 and v2 degrees of freedom 4.2. Probability Density Function. The density of the F distribution is f (F; v1, v2) F>0 (24) 0 otherwise Tabulations of the distribution of F(v1, va) are widely available. Note that Fn, a(Fra m and therefore the critical values can be found from fa vi, vz The following diagram shows the pdf and cdf for the f distribution with FIGURE 8. F Distributtion pdf and cdf Probability Density Function Cumulative Distribution Function 06 2 2 Here is the pdf of the F distribution for some alternative values of pairs of values(vn and v2)

SOME SPECIFIC PROBABILITY DISTRIBUTIONS 9 3.3. Moments of Student’s t-distribution. M ean (t(ν)) = 0 V ar (t(ν)) = ν ν − 2 (22) 4. The F (Fisher variance ratio) statistic 4.1. Distribution Function. If χ2 1(ν1) and χ2 2(ν2) are independently distributed chi-square vari￾ates, then F(ν1, ν2 ) = χ2 1(ν1) ν1 χ2 2(ν2) ν2 = ν2 ν1 · χ2 1(ν1) χ2 2(ν2) (23) has the F density with ν1 and ν2 degrees of freedom. 4.2. Probability Density Function. The density of the F distribution is f ( F; ν1, ν2) = Γ ( ν1+ν2 2 ) Γ ( ν1 2 ) Γ ( ν2 2 ) · ν1 ν2  ν1 2 · F ν1 2 −1 ·  1 + ν1 ν2 F  −(ν1+ν2) 2 F > 0 = 0 otherwise (24) Tabulations of the distribution of F(ν1,ν2) are widely available. Note that Fν1, ν2 ∼  1 F ν2, ν1  and therefore the critical values can be found from fα ν1 , ν2 =  1 f1−α ν2, ν1  . The following diagram shows the pdf and cdf for the F distribution with parameters ν1 = 12 and ν2 = 20. Figure 8. F Distributtion pdf and cdf 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability Density Function X f(X) 0 2 4 6 0 0.2 0.4 0.6 0.8 1 Cumulative Distribution Function X F(X) Here is the pdf of the F distribution for some alternative values of pairs of values (ν1 and ν2)

SOME SPECIFIC PROBABILITY DISTRIBUTIONS FIGURE 9. Probability of Intervals 0.8 2,v2=50) 0.6 1=12 10 0.4 1=6 30 0.2 2 3 4 5 4. 3. moments of the f distribution E(F)= n2(1+v2-2) 6)

10 SOME SPECIFIC PROBABILITY DISTRIBUTIONS Figure 9. Probability of Intervals -1 1 2 3 4 5 6 0.2 0.4 0.6 0.8 fHxL HΝ1 = 12, Ν2 = 50L HΝ1 = 12, Ν2 = 10L HΝ1 = 6, Ν2 = 30L 4.3. moments of the F distribution. E(F) = ν2 ν2 − 2 (25) V ar(F) = 2ν2 2(ν1 + ν2 − 2) ν1(ν2 − 2)2(ν2 − 4) (26)