SOA - Exam P questions and answers 100% verified.

SOA - Exam P questions and answers 100% verified. Chapter j0 Topics jmentioned jin jChapter j0 jthat jthese jflash jcards jdo jnot jcover: j- j jcorrect janswer. j j j- jgraphing jinequalities j - jpiecewise jfunctions j - jone jto jone jfunctions j - jlimits jand jcontinuity - jbasic jrules jof jdifferentiation j - jbasic jintegration - jmethod jof jsubstitution Chapter j0 For jany jtwo jsets jA jand jB, j(A∩B)∪(A∩B') j= j- j jcorrect janswer. j j jA Chapter j0 Two jsets jA jand jB jare jdisjoint jif jA∩B j= j- j jcorrect janswer. j j j∅ Chapter j0 j n(S) jis jdefined jto jbe j- j jcorrect janswer. j j jthe jnumber jof jelements jin ja jset Chapter j0 n(A∩B) j+ jn(A∩B') j= j- j jcorrect janswer. j j jn(A) Chapter j0 In jorder jto jaccount jfor jdouble jcounting, jn(A∪B) j= j- j jcorrect janswer. j j jn(A) j+ jn(B) j- jn(A∩B) Chapter j0 In jorder jto jaccount jfor jdouble jcounting jin jthree jsets, jn(A∪B∪C) j= j- j jcorrect janswer. j j jn(A) j+ jn(B) j+ jn(C) j- jn(A∩B) j- jn(A∩C) j- jn(B∩C) j+ jn(A∩B∩C) j (how jdoes jthis jwork jfor ja jnumber jof jsets jgreater jthan jthree?) Chapter j0 j The jinverse jof ja jfunction jƒ(x) j= jy jis j- j jcorrect janswer. j j jthe jfunction jsolved jfor jx jin jterms jof jy, jsuch jthat jif jƒ(x₀) j= jy₀, jƒ⁻¹(y₀) j= jx₀ Chapter j0 A jquadratic jfunction jof jthe jform jax² j+ jbx j+ jc j= j0 jcan jbe jsolved jwith jthe jquadratic jequation: j- j jcorrect janswer. j j j[-b j± j√(b² j- j4ac)] j/ j2a Chapter j0 j y j= jb^x j↔ jlog.b(y) j= j- j jcorrect janswer. j j jx Chapter j0 j The jnatural jlogarithm jis j- j jcorrect janswer. j j jlog.e(y) j= jln(y) Chapter j0 Important jproperties jof jlogarithms: j- j jcorrect janswer. j j j... Chapter j0 j Partial jdifferentiation jwith jrespect jto jx jis jfound jby j- j jcorrect janswer. j j jdifferentiating jwith jrespect jto jx jand jregarding jy jas ja jconstant, jthen jsubstituting jin jx₀ jand jy₀ Chapter j0 Antiderivatives jof jfrequently jused jfunctions: j- j jcorrect janswer. j j j(for jindividual jflash jcards, jsee jother jdeck) Chapter j0 Useful jintegration jrules: j- j jcorrect janswer. j j j(for jindividual jflash jcards, jsee jother jdeck) Chapter j0 Integration jby jparts: j- j jcorrect janswer. j j j∫ jv j× jdu j= jv j× ju j- j∫ jdv j× ju Chapter j0 ∫ je^(ax) j= j- j jcorrect janswer. j j j[axe^(ax) j- je^(ax)] j/ ja^2 Chapter j0 j ∫ jxe^(ax) j= j- j jcorrect janswer. j j jxe^(ax) j/ ja j- je^(ax) j/ ja^2 Chapter j0 Geometric jprogression j: ja, jar, jar², jar³, j... Sum jof jfirst jn jterms: j- j jcorrect janswer. j j ja j+ jar j+ jar² j+ j... j+ jarⁿ⁻¹ j= ja[1 j+ jr j+ jr² j+ j... j+ jrⁿ⁻¹] j= ja j× j(rⁿ-1)/(r-1) j= ja j× j(1- jrⁿ)/(1-r) Chapter j0 ∫ jxⁿe^(-cx) j= j- j jcorrect janswer. j j jn!/c^(n+1) Chapter j0 Infinite jsum jof jgeometric jseries: j- j jcorrect janswer. j j ja/(1-r) Chapter j0 Arithmetic jprogression: ja, ja j+d, ja j+ j2d, ja j+ j3d, j..., j sum jof jfirst jn jterms j- j jcorrect janswer. j j jna j+ jd j× jn(n-1)/2 Chapter j1 Topics jmentioned jin jchapter j1 jthat jthese jflash jcards jdo jnot jcover: j- j jcorrect janswer. j j jDefinitions: j - jevent - junion jof jevents - jintersection jof jevents - jcomplement - jcontinuous jprobability jspace Chapter j1 Sample jpoint j- j jcorrect janswer. j j jthe jsimple joutcome jof ja jrandom jexperiment Chapter j1 Probability jspace j- j jcorrect janswer. j j jthe jcollection jof jall jpossible jsample jpoints jrelated jto ja jspecific jexperiment Chapter j1 Mutually jexclusive joutcomes j- j jcorrect janswer. j j jcannot joccur jsimultaneously j(disjoint) A∩B j= j∅ Chapter j1 Exhaustive joutcomes j- j jcorrect janswer. j j joutcomes jthat jcombine jto jbe jthe jentire jprobability jspace, jor jequivalently, jat jleast jone jof jthe joutcomes jmust joccur jwhenever jthe jexperiment jis jperformed A₁∪A₂∪...∪A.n j= jS, jthe jentire jprobability jspace Chapter j1 Subevent j(subset) j- j jcorrect janswer. j j jB jcontains jall jsample jpoints jin jevent jA, jthen jA jis jsubevent jof jB, jA⊂B. jThe joccurrence jof jA jimplies jevent jB. Chapter j1 j Partition jof jA j- j jcorrect janswer. j j jC₁, jC₂, jC₃₃, j..., jC.n jform ja jpartition jof jA jif jA j= jthe junion jof jall jCs jand jthe jCs jare jmutually jexclusive Chapter j1 DeMorgan's jLaws: (A∪B)' j= (A∩B)' j= j- j jcorrect janswer. j j j(A∪B)' j= jA'∪B' (A∩B)' j= jA'∩B' Chapter j1 Indicator jfunction jfor jevent jA, jI.A(x) j= j- j jcorrect janswer. j j jI.A(x) j= j1 jif jx j∈ jA I.A(x) j= j0 jif jx j∉ jA Chapter j1 A∩(B₁∪B₂∪...∪B.n) j= j A∪(B₁∩B₂∩...∩B.n) j= j- j jcorrect janswer. j j j(A∩B₁)∪...∪(A∩B.n) (A∪B₁)∩...∩(A∪B.n) Chapter j1 If jB₁...B.n jare jexhaustive jevents jthen jfor jany jevent jA, jA j= j- j jcorrect janswer. j j j(A∩B₁)∪...∪(A∩B.n) Chapter j1 If jB₁...B.n jare jexhaustive jevents jand jmutually jexclusive jthen jthey jform ja j_____ jof jthe jprobability jspace. j- j jcorrect janswer. j j jpartition Chapter j1 For jany jevents jA jand jB, j (A∩B)∪(A∩B') j= jA∩(B∪B') j= j- j jcorrect janswer. j j jA Chapter j1 For jany jevent jA, jA∪A' j= j A∩A' j= j- j jcorrect janswer. j j jS ∅ Chapter j1 A j- jB j= j- j jcorrect janswer. j j jA∩B' Chapter j1 If jA⊂B jthen jA∪B j= j and jA∩B j= j- j jcorrect janswer. j j jB A Chapter j1 Probability jfunction jof ja jdiscrete jprobability jspace jmust jsatisfy jwhat jtwo jconditions? j- j jcorrect janswer. j j j(i) j0 j≤ jP[a.i] j≤ j1 jfor jeach ja.i jin jthe jsample jspace (ii) j∑P[a.i] j= j1 Chapter j1 Uniform jprobability jfunction j- j jcorrect janswer. j j jfinite jnumber jof jsample jpoints, jk, jeach jof jwhich jhave jthe jsame jprobability jof joccurring j(1/k) Chapter j1 In ja jdiscrete jprobability jspace, jevent jA jconsists jof ja jsubset jof jevents j(a.i), jthen j∑P[a.i] jfor jall ja.i jin jA j= j- j jcorrect janswer. j j jP[A] Chapter j1 Basic jprobability jrules: - jP[S] j= j - jP[∅] j= - jfor jdisjoint jevents jA₁...A.n, jP[∪A.i jfrom ji j= j1 jto jn] j= j - jIf jA j⊂ jB, jthen jP[A] j - jP(A∪B) j= j - jFor jtwo jevents jA jand jB, jP[A] j= - jP(A∪B∪C) j= j- j jcorrect janswer. j j j- j1 - j0 - j∑P[A.i] - jP[A] j≤ jP[B] - jP(A) j+ jP(B) j- jP[A∩B] - jP[A∩B'] j+ jP[A∩B] - jP(A) j+ jP(B) j+ jP(C) j- jP(A∩B) j- jP(A∩C) j- jP(B∩C) j+ jP(A∩B∩C) Chapter j1 Basic jprobability jrules jcon't: j - jP[A'] j= j - jFor jmutually jexclusive jexhaustive jevents jB₁...B.n, jP[A] j= j - jIf jevent jA jconsists jof jm jpoints jof ja juniform jprobability jfunction jon ja jprobability jspace jwith jk jpoints, jP[A] j= j - jPercentage jand jproportion jare jsynonymous jwith jprobability - jFor jany jevents jA₁...A.n, jP[∪Ai] j≤ j∑P[A.i] jwith jthe jequality jholding jif jand jonly jif jthe jevents jare j_____ j- j jcorrect janswer. j j j- j1 j- jP[A] - j∑P[A∩B.i jfrom ji j= j1 jto ji j=n] - jm j/ jk - - jmutually jexclusive Chapter j2 Items jin jChapter j2 jthat jthese jflash jcards jdo jnot jcover: j- j jcorrect janswer. j j jDefinitions/concepts: - jconditional jprobability j - jevent jtree - jprior jprobabilities j - jposterior jprobabilities Chapter j2 P[B|A] j= j- j jcorrect janswer. j j jP[B|A] j= jP[B∩A] j/ jP[A] Chapter j2 P[B∩A] j= j- j jcorrect janswer. j j jP[B∩A] j= jP[B|A] j× jP[A] Chapter j2 Baye's jrule j- j jcorrect janswer. j j jP[A|B] j= jP[A∩B] j/ jP[B] j= j(P[B|A] j× jP[A]) j/ j(P[B|A] j× jP[A] j+ jP[B|A'] j× jP[A']) Chapter j2 implications jof jBaye's jtheorem: - jP[B] j= j - jP[B'|A] j= j1 j- jP[B|A] - jP[B'|A'] j= j- j jcorrect janswer. j j j- jP[B] j= jP[B∩A] j+ jP[B∩A'] - jP[B'|A] j= j1 j- jP[B|A] - jP[B'|A'] j= j1- jP[B|A'] Chapter j2 Baye's jrule j(extended) j if jevents jA₁ j... jA.n jform ja jpartition jof jthe jentire jprobability jspace, jP[A.j|B] j= jP[B∩A.j] j/ jP[B] j= j- j jcorrect janswer. j j jP[A.j|B] j= jP[B∩A.j] j/ jP[B] j= jP[B∩A.j] j/ j∑P[B∩A.i] j= jP[B|A.j] j× jP[A.j] j/ j∑(P[B|A.i] j× jP[A.i]) Chapter j2 - jThe jgeneral jLaw jof jTotal jProbability j- j jcorrect janswer. j j j- jP[B] j= j∑(P[B|A.i] j× jP[A.i]) Chapter j2 What jrelationships jdo jindependent jevents jA jand jB jsatisfy? j- j jcorrect janswer. j j jP[A∩B] j= jP[A] j× jP[B] P[A|B] j= jP[A] j(and jvice jversa) Chapter j2 Rules jconcerning jconditional jprobability jand jindependence: (i) jP[A∩B] j= j (ii) jIf jP[A₁∩A₂∩...∩A.n-1] j j0, jthen jP[A₁∩...∩A.n] j= j (iii) jP[A'|B] j= j (iv) jP[A∪B|C] j= j (v) jIf jA j⊂ jB, jthen jP[A|B] j= j (vi) jIf jA jand jB jare jindependent, jwhat jdoes jthis jimply jabout jA' jand jB' jand jA jand jB'? (vii) j∅ jis j____ jof jany jevent jA j- j jcorrect janswer. j j j(i) jP[A∩B] j= jP[B|A] j× jP[A] (ii) jP[A₁∩...∩A.n] j= jP[A₁] j× jP[A₂|A₁] j× jP[A₃|A₁∩A₂] j× j... j× jP[A.n|A₁∩...∩A.n j-1) (iii) jP[A'|B] j= j1- jP[A|B] (iv) jP[A∪B|C] j= jP[A|C] j+ jP[B|C] j- jP[A∩B|C] (v) jP[A|B] j= jP[A∩B] j/ jP[B] j= jP[A] j/ jP[B] (vi) jA jand jB' jare jindependent jand jA' jand jB' jare jindependent (vii) jindependent j(for jproof jsee jpage j65) Chapter j3 The jnumber jof jways jn jobjects jcan jbe jordered jor jpermuted jis j- j jcorrect janswer. j j jn! Chapter j3 The jnumber jof jways jyou jcan jchoose jan jordered jsubset jof jsize jk jwithout jreplacement jfrom ja jcollection jof jn jobjects jis j- j jcorrect janswer. j j jn! j/ j(n j- jk)! Chapter j3 Given jn jobjects, jn₁ jare jtype j1, jn₂ jare jtype j2, j..., jn.n jare jtype jn, jthe jnumber jof jways jto jorder jthem jis j- j jcorrect janswer. j j jn! j/ j(n₁! j× jn₂! j× j... j× jn.n!) Chapter j3 Given jn jobjects, jthe jnumber jof jways jto jchoose ja jsubset jk jwithout jreplacement jand jwithout jregard jto jthe jorder j(combination jof jn jobjects) jis j This jis jdenoted jas j_____, jand jis jreferred jto jas jthe j______ j- j jcorrect janswer. j j jn! j/ j(k! j× j(n j- jk)!) C(n,k) j(n jchoose jk), jbinomial jcoefficient Chapter j3 C(n, j0) C(n, jn) C(n, j1) j- j jcorrect janswer. j j j1 1 n Chapter j3 Binomial jTheorem In jthe jpower jseries jexpansion jof j(1+t)ⁿ jthe jcoefficient jof jt^k jis Thus, j(1+t)ⁿ j= j- j jcorrect janswer. j j jC(n, jk) (1+t)ⁿ j= j∑C(n, jk) j× jt^k j= j1 j+ jnt j+ jn(n j- j1) j/ j2 j× jt^2 j+ jn(n-1)(n-2) j/ j6 j× jt^3 j... j If jn jis jan jinteger jthen jthe jsummation jstops jat jk j= jn jand jthe jseries jis jvalid jfor jany jreal jnumber jt, jbut jif jN jis jnot jan jinteger jthen jthe jseries jis jvalid jif jabs(t) j j1 Chapter j3 Binomial jTheorem jcont'd Given jn jobjects, jn₁ jare jtype j1, jn₂ jare jtype j2, j..., jn.s jare jtype jn, jthe jnumber jof jways jof jchoosing ja jsubset jof jsize jk jwith jk₁ jobjects jof jtype j1 jand jk.s jobjects jtype jn jis j- j jcorrect janswer. j j jC(n₁, jk₁) j× jC(n₂, jk₂) j× j... j× jC(n.s, jk.s) Chapter j3 Multinomial jtheorem j(general jbinomial jtheorem) In jthe jpower jseries jexpansion jof j(t₁ j+ jt₂ j+ j... j+ jt.s)ⁿ, jthe jcoefficient jof jt₁^k₁ j× j... j× jt.s^k.s jwhere jk₁ j+ j... j+ jk.s j= jn jis j- j jcorrect janswer. j j jC(n, jk₁ jk₂ j... jk.s) j= jn! j/ j(k₁! j× j... j× jk.s!) for jexample jthe jcoefficient jof jxy^2 jof jthe jexpansion jof j(1 j+ jx j+ jy)^4 j= jC(4, j1 j1 j2) Chapter j4 What jis ja jrandom jvariable? j- j jcorrect janswer. j j jA jrandom jvariable jis jfunction jon ja jprobability jspace jS. jThis jfunction jassigns ja jreal jnumber jX9s) jto jeach jsample jpoint js j∈ jS. jLess jformally, ja jrandom jvariable jconsists jof jthe jpossible jvalues jthat jcan joccur jand jthe jprobabilities jof jthose jvalues. Chapter j4 What jis ja jdiscrete jrandom jvariable? j- j jcorrect janswer. j j jA jrandom jvariable jX jis jsaid jto jbe jdiscrete jif jit jtakes jon jonly jvalues jfrom ja jfinite jor jcountable jinfinite jsequence j(usually jthe jintegers jor jsome jset jof jintegers). Chapter j4 What jproperties jmust jthe jprobability jfunction jp(x) jsatisfy jfor ja jdiscrete jrandom jvariable? j- j jcorrect janswer. j j j(i) j0 j≤ jp(x) j≤ j1 jfor jeach ja.i jin jthe jsample jspace (ii) j∑p(x) j= j1 Chapter j4 The jprobability jfunction jp(x) jcan jbe jdescribed jby ja jprobability jplot jor jhistogram. jFor jinstance, jif j2 jhas ja j.5 jchance jof joccurring, jp(2) j= j- j jcorrect janswer. j j j.5 Chapter j4 Continuous jrandom jvariable j- j jcorrect janswer. j j jA jrandom jvariable jwhich jcan jassume jnumerical jvalues jfrom jan jinterval jof jreal jnumbers. jThe jprobability jspace jis jthis jinterval Chapter j4 Probability jdensity jfunction j(pdf) j, jdenoted jf(x) jmust jsatisfy jwhat jtwo jproperties? P[a j jX j jb] j= j(in jterms jof jf(x)) For ja jcontinuous jrandom jvariable jP(X j= jc) j= j- j jcorrect janswer. j j j(i) jf(x) j≥ j0 jfor jall jx (ii) j∫f(x) jfrom j-∞ jto j∞ j= j1 P[a j jX j jb] j= j∫f(x) jfrom ja jto jb P(X j= jc) j= j0 Chapter j4 Mixed jdistribution j- j jcorrect janswer. j j jIf ja jrandom jvariable jhas jpoints jwith ja jnonzero jprobability jmass jcombined jwith ja jcontinuous jpdf jon jone jor jmore jintervals, jit jhas ja jmixed jdistribution. jThe jprobabilites jof jthe jpoint(s) jand jthe jpdf(s) jmust jsum jto j1. Chapter j4 Cumulative jdistribution jfunction j- j jcorrect janswer. j j jF(x) j= jP[X j≤ jx] F(x) j= j∫f(t) jfrom j-inf jto jx P[a j jX j≤ jb] j= jF9b) j- jF9a) Chapter j4 Survival jfunction tail j- j jcorrect janswer. j j jS(x) j= j1 j- jF(x) j= jP[X j jx] The jevent jX j jx jis jreferred jto jas ja jtail j(or jright jtail) jof jthe jdistribution Chapter j4