Economics 520, Fall 2010Lecture Note 6: Special Distributions continued (CB 3.3-3.4)1. Gamma Distribution The Gamma distribution with parameters α > 0 and β > 0 isfX(x) =xα−1e−xβΓ(α)βα,for x > 0 and 0 elsewhere. Recall that the gamma function is defined asΓ(α) =Z∞0tα−1e−tdt.It has the properties thatΓ(α + 1) = α · Γ(α), Γ(1) =Z∞0e−tdt = 1, Γ(1/2) =√π,with the first two implying thatΓ(n) = (n − 1)!for integer n. Setting β = 1/λ and α = 1 we get the exponential distribution with arrival rateλ, so the exponential distribution is a special case of the Gamma.The moment-generating function of the Gamma distribution isMX(t) =1(1 − βt)α,and the cumulant generating function isKX(t) = −α ln(1 − βt).The mean and variance are αβ and αβ2respectively.The Gamma distribution can be motivated in a similar way to the waiting-time motivationof the exponential distribution. In the previous lecture note we showed that the exponentialdistribution can be thought of as the waiting time until the first occurence of an event. In thesame way, the Gamma distribution represents the waiting time until the rth event. Let X bethis waiting time. ThenFX(x) = P r(X ≤ x) = 1 − P r(fewer than r events in interval [0, x)).The last probability is equal to the probability that a Poisson random variable with arrival rate1λx is less than r:FX(x) = 1 −r−1Xk=0e−λx(λx)kk!.(Note that if we set r = 1, we have FX(x) = 1 − exp(−λx), the cumulative distributionfunction for the exponential distribution, the waiting time until the first event.) Take thederivative with respect to x to get the probability density function:fX(x) =r−1Xk=0λk+1xke−λxk!−r−1Xk=1λkxk−1e−λx(k −1)!.Note that the second summand is only from k = 1 to r −1, not from k = 0 to r−1. Changingthe second summation from k = 1 to k = r − 1 to the summation from k = 0 to k = r − 2,we can write this asfX(x) =r−1Xk=0λk+1xke−λxk!−r−2Xk=0λk+1xke−λxk!=λr(r −1)!xr−1e−λx.This is a Gamma distribution with parameters α = r and β = 1/λ. When α is not an integer,we cannot use the interpretation as the waiting time until the αth event, but the probabilitydensity is still well defined, and the extra flexibility is often useful for modelling purposes.Since the Gamma distribution can be interpreted as the waiting time for the rth event, thefollowing result makes sense. Suppose that Y1, . . . , Yrare independent exponential randomvariables with parameter λ. Then X =Pri=1Yiis a Gamma random variable with parametersα = r and β = 1/λ. More generally, if Y1, . . . , Yrare independent Gamma random variableswith parameters αiand β, then X =PYiis Gamma with parameters α =Pri=1αiand β.2. Chi-squared Distribution A special case of the Gamma distribution is the Chi-squared distri-bution. Take α = k/2, where k is a positive integer, and β = 2, we have a Chi-squareddistribution with degrees of freedom equal to k. Its pdf isfX(x) =xk/2−1e−x/2Γ(k/2)2k/2,for x positive. Its moment generating function isMX(t) =1(1 − 2t)k/2,and its mean and variance are k and 2k respectively.23. Normal Distribution One of the most important distributions is the normal distribution. It doesnot have as easy a motivation as some of the other distributions, but it is of fundamental impor-tance as an approximation to a large number of statistics through the central limit theorem. A to be discussed infuture lecturesrandom variable X has a normal distribution with parameters µ and σ2, denoted by N(µ, σ2),if its pdf isfX(x) =1√2πσ2exp−(x − µ)22σ2,for −∞ < x < ∞, with the parameter space −∞ < µ < ∞ and σ2> 0. First considerthe moment generating function. Its derivation relies on a trick we have used before, namelyusing the fact that the pdf and pmf respectively integrate and add up to one:MX(t) =Z∞−∞1√2πσ2expxt −(x − µ)22σ2dx=Z∞−∞1√2πσ2exp−(x − µ)2− 2σ2xt2σ2dx=Z∞−∞1√2πσ2expt2σ2/2 + µt −(x − µ − σ2t)22σ2dx= exp(µt + σ2t2/2) ·Z∞−∞1√2πσ2exp−(x − µ − σ2t)22σ2dx= exp(µt + σ2t2/2).The cumulant generating function is KX(t) = µt + σ2t2/2 and hence the mean is µ and thevariance σ2.One of the most important properties of the normal distribution is that linear transformationsof normal random variables are also normally distributed. Consider a random variable Xwith a N(µ, σ2) distribution, and consider the transformation Y = a + bX. Then, throughthe moment generating function,MY(t) = E(etY) = E(e(a+bX)t) = eat· E(e(bt)X)= eta· MX(bt) = exp(at + µbt + σ2b2t2/2) = exp((a + bµ)t + b2σ2t2/2)= exp(˜µt + ˜σ2t2/2).Hence Y has a normal distribution with mean ˜µ = a + bµ and variance ˜σ2= b2σ2. Aparticularly useful transformation is that from X to Y = (X −µ)/σ. Then Y ∼ N(0, 1), thestandard normal distribution with mean zero and unit variance.Finally, there is a close connection between the normal distribution and the Chi-squared dis-tribution. If X has a standard normal distribution N(0, 1), then Y = X2has a Chi-squared3distribution with degrees of freedom equal to one. One argument goes as follows: The mo-ment generating function of Y isMY(t) = E(etY) = E(etX2) =Z∞−∞1√2πexp−12x2+ tx2dx=Z∞−∞1√2πexp−12/(1 + 2t)x2dx=p1/(1 − 2t)Z∞−∞1p2π/(1 − 2t)exp−12/(1 − 2t)x2dx =1(1 − 2t)1/2.This is the mgf for a Chi-squared distribution with degrees of freedom equal to one.4. Cauchy Distribution A random variable X has a Cauchy distribution centered around θ if ithas pdffX(x) =1π·11 + (x − θ)2,for −∞ < x < ∞. This distribution has no moments. The median and mode are bothequal to θ. The pdf looks very similar to the pdf for the normal distribution but it has thickertails. It is often used for modelling variables with high kurtosis, that is, which infrequentlytake on extremely large values, such as stock prices. An interesting property of the Cauchydistribution is that if the sequence of independent random variables X1, X2, . . . , Xnall havethe same Cauchy distribution centered around θ, then the average¯X =Pni=1Xi/n also hasthat same Cauchy distribution centered around θ. In other words, laws of large numbers willbe seen not to apply to distributions like Cauchy distributions.5. Beta Distribution. Suppose two independent random variables Y1and Y2have Gamma distri-butions with