Continuous Random Variables: The GammaDensity,Uniform density, Density of functions of arandom variableJuana [email protected] Department of StatisticsJuly 17, 2013J. Sanchez Continuous Random VariablesMore families of continuous random variablesI. The GammaII. The Gamma random variable and its pdf.III. The Uniform random variable.IV. Density of functions of a continuous random variable.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 VariablesNotesNotesNotesNotesContinuing integrating by parts all new Gamma functions that appearuntil the end, we get:Γ(n) = (n − 1)Γ(n − 1) = (n − 1)(n − 2)Γ(n − 2)= (n − 1)(n − 2) · · · Γ(1) = (n − 1)! n > 0Notice thatΓ(1) =Ze−xdx = 1Γ(α + 1) = αΓ(α)J. Sanchez Continuous Random VariablesII. 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 X is an exponential random variable.If α = n/2 and λ = 1/2 then X is a Chi-square random variable withn degrees of freedom.J. Sanchez Continuous Random VariablesExpected Value and Variance of a Gamma Random variableExpected value of Gamma r.v.E (X ) =αλThe proof is left as an exercise in your next homework. Using thedefinition, you can show that the above is true. The Gamma function willappear in your proof.Variance 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 VariablesNotesNotesNotesNotesExercise for next dayIf a random variable Z is N(0,1), what will be the distribution of Z2?Show using moment generating functions.J. 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 = 1/2f (x) =(λe−λx(λx)α−1Γ(α)0x ≥ 0, α > 0, λ > 0x < 0Final formula for f(x)?f (x) =xe−x24x ≥ 0J. 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 = 1/2f (x) =(λe−λx(λx)α−1Γ(α)0x ≥ 0, α > 0, λ > 0x < 0Final formula for f(x)?f (x) =xe−x24x ≥ 0J. 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 = 1/2f (x) =(λe−λx(λx)α−1Γ(α)0x ≥ 0, α > 0, λ > 0x < 0Final formula for f(x)?f (x) =xe−x24x ≥ 0J. Sanchez Continuous Random VariablesNotesNotesNotesNotesExampleThe 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 = 1/2f (x) =(λe−λx(λx)α−1Γ(α)0x ≥ 0, α > 0, λ > 0x < 0Final formula for f(x)?f (x) =xe−x24x ≥ 0J. Sanchez Continuous Random VariablesExercise for next dayFour-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 VariablesIII. The Uniform random variableExperiments that consist of observing events in a certain time frame,such as buses arriving at a bus stop or telephone calls coming into aswitchboard during a specified period, are sometimes modeled with aUniform r.v. Suppose that we know that one such event has occurred inthe time interval α, β : a bus arrived between 8 and 8:10. It may then beof interest to place a probability density on the actual time of occurrenceof the event under observation, which would be the r.v. X. The Uniformmodel assumes that X is equally likely to lie in any small subinterval.J. Sanchez Continuous Random VariablesPdf and CDF of a Uniform random variableA uniform random variable X with parameters α and β can becharacterized as follows:Density function (pdf)f (x) =1β−αα < x < β0 otherwiseCumulative Distribution Function (CDF)F (x ) = Prob(X ≤ x) =Rxα1β−αdx=x−αβ−αα < x < β,= 0 otherwiseand F0(x) =1β−α= f (x)We can use the cdf to compute probabilities of exponential randomvariablesP(a < X < b) = F (b) − F (a)J. Sanchez Continuous Random VariablesNotesNotesNotesNotesFigure: Uniform pdf (α = 1, β = 4)Figure: CDF of auniform(α = 1, β = 4)J. Sanchez Continuous Random VariablesFinding probabilities for Uniform random variablesProbabilitiesP(a < x < b) =Zba1β − αdx = (b − a)1β − αIf a = α, b = β then the integral is 1.You may play with several values of a and b. But notice also that we canuse the cumulative distribution function to compute probabilitiesP(a < x < b) = F (b) − F (a)J. Sanchez Continuous Random VariablesExampleLet X be a uniformly distributed random variable over (0,10). Calculatethe following probabilities(a)P(X < 3) =Z30110dx = 3/10(b)P(X > 6) =Z106110dx = 4/10(c)P(3 < X < 8) =Z83110dx = 1/2J. Sanchez Continuous Random VariablesExampleA bomb is to be dropped along a 1-mile-long line that stretches across apractice target zone. The target zones center is at the midpoint of theline. The target will be destroyed if the bomb falls within 1/10 mile oneither side of the center. Find the probability that the target will bedestroyed, given that the bomb falls at a random location along the line.X ∼ U(−0.5, 0.5) reads as ” let Xbe Uniform in the interval−0.5, 0.5. Thusf (x) = 1 −
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