probability homework

PDF “density” 1 2 2 𝑓(𝑥) = 𝑒 −(𝑥−µ) /2𝜎 𝜎√2𝜋 Normal MGF NOTES Is very desirable to be used in applications, but both the 1 2 2 𝑒 µ𝑡+2𝜎 𝑡 ALT/ Notation N(µ,σ) population mean = µ and population st dev = σ Sample space Ω is that x is defined for all values (-∞ < x < ∞ ) T dist must be known. Is an approximation to the normal dist for smaller samples, with degrees freedom = v. N/A ALT/ Notation T v PLEASE LET ME KNOW IF YOU FIND AN MGF ☺ Sample space Ω is that x is defined for only (-∞ < x < ∞ ) Chi square 1 𝑣 𝑣 22 𝛤 ( 2 ) (1 − 2𝑡)−𝑣/2 Is essentially the T dist squared, which is usefull in applications for things like mean squared error analysis 𝑎 𝑎−𝑡 Textbooks use for good easy example – technically is time decay probability 1 𝑡𝐿 (𝑒 − 1) 𝑡𝐿 Textbooks use for good easy example – technically equal likely 𝑣 𝑥 𝑥 2−1 𝑒 −2 ALT/ Notation χ v Sample space Ω is that is defined for x > 0 only Exponential 𝑎𝑒 −𝑎𝑥 if x >0 Sample space Ω is that x is defined for x > 0 only. 1 𝑖𝑓 0 < 𝑥 < 𝐿, 𝑏𝑢𝑡 𝐿 = 0 𝑒𝑙𝑠𝑒 Uniform 𝑑𝑀 ɸ NOTE: the nth moment can be found as µ𝑀 = 𝑑𝑡 𝑀 (0) where ɸ is the MGF. NOTE: using the moments on can find E(x) = µ1 ∞ 1 & VAR(x) = µ2 − (µ1 )2 𝛤(𝑛) = ∫0 𝑥 𝑛−1 𝑒 −𝑥 𝑑𝑥 but use: 𝛤(𝑛) = (𝑛 − 1)! or 𝛤 (2 + 𝑛) = (2𝑛−1)‼ 2𝑛 *  where n is integer. MA 412 Paper HW 9 Name: ___________________________ 1. Compute 𝐸(𝑥) for 𝑁~(3,1) {e.g. a normal with mean 𝜇 = 3 and st dev 𝜎 = 1} a. by integral definition (set up & simplify integral and use integration software) 𝐸(𝑥) = _________________ b. by MGF 𝐸(𝑥) = _________________ 2. Compute 𝑃(0 < 𝑥 < 1) for T2 (set up & simplify integral and use integration software) 𝑃(0 < 𝑥 < 1) = ___________ 3. Compute 𝐸(𝑥) for 2e-2x a. by integral definition (do by hand) 𝐸(𝑥) = _________________ b. by MGF 𝐸(𝑥) = _________________ For #4-7, use the chi square PDF with v= 𝟒, i.e. 𝒇(𝒙) = 𝟒 𝟏 𝟐𝟐 𝜞(𝟐) 𝒙 𝒙𝟐−𝟏 𝒆−𝟐 , for 𝒙 > 𝟎 4. Use the traditional definition 𝐸(𝑥) = ∫𝛺 𝑥 ∙ 𝑓(𝑥)𝑑𝑥 to compute the expectation (do by hand (IBP)) 𝐸(𝑥) = _________________ 5. Use table provided to write MGF for this PDF & use it to compute the 1st & 2nd moments. MGF = _________________ 𝜇1 = _________ 𝜇2 = ________ 6. Compute the MEAN µ = 𝜇1 and verify that it yields the same solution as Q4. µ = _________________ 6. 7. Compute the variance as 𝜎 2 = 𝜇2 − (𝜇)2 and verify that it yields the same solution as #5. 𝜎 2 = _________________ 𝟏 For #8-9, use the uniform PDF with 𝑳 = 𝟑, i.e. 𝒇(𝒙) = , for 𝟎 < 𝒙 < 𝟑 𝟑 8. Use the traditional definition µ = ∫𝛺 𝑥 ∙ 𝑓(𝑥)𝑑𝑥 to compute the mean (can do by hand) µ = _________________ 9. Use table provided to write MGF for this PDF & use it to compute the 1st moment Hint: you will need to expand 𝑒 3𝑡 using a Taylor series. 𝑥𝑛 1 1 2 3 Hint: Taylor series for 𝑒 𝑥 = ∑∞ 𝑛=0 ( 𝑛! ) = 1 + 𝑥 + 2! 𝑥 + 3! 𝑥 + ⋯ MGF ( as defined ) = _________________ MGF ( rewritten with Taylor’s ) = _________________ 𝜇1 = _________