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 < x = 0 otherwise (10) where Γ ( · ) is the gamma function defined by Γ (r ) = R ∞ 0 u r − 1 e −u du r > 0 (11) Note that for positive integer values of r, Γ(r) = (r - 1)!