Continuous Random Variables: The GammaDensity.Juana [email protected] Department of StatisticsJ. Sanchez Continuous Random VariablesMore families of continuous random variablesToday’s lecture has material from Ross, chapter 5I. The Gamma function from CalculusII. The Gamma random variable and its pdf.J. Sanchez Continuous Random VariablesPhenomena that are gammaSummer rainfall totalsLifetime of electronic components.J. Sanchez Continuous Random VariablesI. The GammaGamma IntegralΓ(α) =R∞0e−yyα−1dyRecall integration by parts:Rudv = uv | −Rvdumake:u = yα−1; dv = e−ydy;Γ(α) = −e−yyα−1∞0+R∞0e−y(α − 1)yα−2dy= (α − 1)R∞0e−yyα−2dy= (α − 1)Γ(α − 1)J. Sanchez Continuous Random VariablesContinuing integrating by parts all new Gamma functions that appearuntil the end, we get:Γ(α) = (α − 1)Γ(α − 1) = (α − 1)(α − 2)Γ(α − 2)= (α − 1)(α − 2) · · · Γ(1) = (α − 1)! α > 0Notice thatΓ(1) =Ze−xdx = 1Γ(α + 1) = αΓ(α)J. Sanchez Continuous Random VariablesI I. The Gamma random variable and its pdfA random variable X ≥ 0 is Gamma distributed with parameters α and λiff (x) =(λe−λx(λx )α−1Γ(α)0x ≥ 0, α > 0, λ > 0x < 0Notice that:If α = 1, then what kind of random variable is X ?If α = n/2 and λ = 1/2 then X is a Chi-square random variable withn degrees of freedom.J. Sanchez Continuous Random VariablesExp ected Value and Variance of a Gamma Random variableExpected value of Gamma r.v.E (X ) =αλVariance of Gamma r.v.Var (X ) =αλ2J. Sanchez Continuous Random VariablesMoment generating function of the Gamma randomvariableM(t) =λλ − tα=11 − t/λαJ. Sanchez Continuous Random VariablesExampleThe response times at an on-line computer terminal have, approximately,a gamma distribution, with a mean of 4 seconds and a variance of 8.Write the probability density function for these timesE (X ) =αλ= 4 ⇒ λ =α4VarX =αλ2=16αα2=16α= 8 → α = 2; λ = 1/2Final formula for f(x)? Write it !J. Sanchez Continuous Random VariablesExample -complete it !Four-week summer rainfall totals (in inches) in a certain section of theMidwestern United States follow a gamma distribution with parametersα = 1.6 and λ = 2. On average how much is the four-week rainfall total?What is a typical deviation from that average on any day?J. Sanchez Continuous Random
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