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 = _________

Collepals.com Plagiarism Free Papers

Are you looking for custom essay writing service or even dissertation writing services? Just request for our write my paper service, and we'll match you with the best essay writer in your subject! With an exceptional team of professional academic experts in a wide range of subjects, we can guarantee you an unrivaled quality of custom-written papers.

Get ZERO PLAGIARISM, HUMAN WRITTEN ESSAYS

Why Hire Collepals.com writers to do your paper?

Quality- We are experienced and have access to ample research materials.

We write plagiarism Free Content

Confidential- We never share or sell your personal information to third parties.

Support-Chat with us today! We are always waiting to answer all your questions.