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373 Use the comparison theorem to determine whether a definite integral is convergent0 1 3 2 9 8 3 7 3 0 6 / 2 3 3 5 3 4 3 , 2 1 2 9 0 0 7 2 1 ;X(t) = E etX If M X(t) = M Y(t) for all tin an interval around 0 then X =d Y The moment generating function can be used to \generate all the moments of a distribution, ie we can take derivatives of the mgf with respect to tand evaluate at t= 0, ie we have that M(n) X (t)j t=0 = E(X n) 3

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0, eq (S911) can be rewritten as X(w) = e(/ 2w)t dt 2 1 j2w It is convenient to write X(o) in terms of its real and imaginary parts X(w) 2 1j2 2 j4w 1 j2w 1 j2wJ 1 4W2 20 / 2 3 = 8 0 <Z e ax2 dx= p ˇ 2 p a erf x p a (69) Z xe ax2 dx= 1 2a e 2 (70) Z x2e ax2 dx= 1 4 r ˇ a3 erf(x p a) x 2a e ax2 Integrals with Trigonometric Functions (71) Z sinaxdx= 1 a cosax (72) Z sin2 axdx= x 2 sin2ax 4a (73) Z sin3 axdx= 3cosax 4a cos3ax 12a (74) Z sinn axdx= 1 a cosax 2F 1 1 2;

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X(t) = EetX At this point in the course we have only considered discrete RV’s We have not yet defined continuous RV’s or their expectation, but when we do the definition of the mgf for a continuous RV will be exactly the same Example Let X be geometric with parameter p Find its mgfWhere the interchange of integrals is justi ed by Fubini’s theorem for improper Riemann integralsV P = power, I or J = Latin influare, international ampere, or intensity and R = resistance V = voltage, electric potential difference Δ V or E = electromotive force (emf = voltage)

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It doesn’t matter if you are moving to Texas or you’ve lived here your entire life, the Lone Star State is fascinating We all know Texas is too awesome to sum up in just 26 words, so we’ve created your Texas AZ list This list provides everything you need to know about Texas from A to Z Get ready — your state pride is about to get˙ (112) Asymptotic Expansion for Large x(x>2) Since erfc x= 2 p ˇ Z 1 x e t2 dt= 2 p ˇ Z 1 x 1 t e t2 tdt we canT D X ΃Z T i P Q X ΁E R U O x X ΁E U m j KSN530S AKSN530D ( m } N Y @ ^ C v)

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2 67 8 ' n 3 4 °R and the power law equation (formula) P = I ×3 y y ' 2 ' X Y FigureS13 2 Themomentgeneratingfunctionofc 1X 1 c 2X 2 is Eet(c 1X 1c 2X 2)=Eetc 1X 1Eetc 2X 2=(1−β 1c 1t) −α 1(1−β 2c 2t) −α 2 Ifβ 1c 1 =β 2c 2,thenX 1 X 2 isgammawithα=α 1 α 2 andβ=β ic i 3 M(t)=Eexp( n i=1 c iX i)= n i=1 Eexp(tc iX i)= n i=1 M i(c it) 4 ApplyProblem3withc i=1foralliThus M Y(t)= n i=1 M i(t)= n i=1 expλi(et−1

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†Heat sources Q(x;t) = heat energy per unit volume generated per unit time † Temperature u(x;t) † Speciflc heat c = the heat energy that must be supplied to a unit mass of a substance to raise its temperature one unit † Mass density ‰(x) = mass per unit volume Conservation of heat energyAbout Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy &The Fourier transform of x(t) is X(w) = x(t)e jw dt = fet/2 u(t)e dt (S911) Since u(t) = 0 for t <

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Z= 4t 1 8(12 points) Using cylindrical coordinates, nd the parametric equations of the curve that is the intersection of the cylinder x 2 y 2 = 4 and the cone z=C PN T e r m i n at o r Z T Fo r us e a a n ad ju sta ble co ro l/ limi te r th e r m o s t a w it h T em o neat i gc ab le sy st e m I P 6 60 °What is the z value such that 52% of the data are to its left?

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) * g ) f ?The exponential function is a mathematical function denoted by () = ⁡ or (where the argument x is written as an exponent)It can be defined in several equivalent waysIts ubiquitous occurrence in pure and applied mathematics has led mathematician W Rudin to opine that the exponential function is the most important function in mathematics Its value at 1, = ⁡ (), is a mathematicalLearning Objectives 371 Evaluate an integral over an infinite interval;

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0 0 The next lemma shows that the tail bounds of Lemma 13 are sufficient to show that the absolute moments of X ∼ subG(σ 2) can be bounded by those of Z ∼ N(0,σ 2) up to multiplicative constants Lemma 14 Let X be a random variable such that t 2 IPX >t ≤ 2exp (− ), 2σ2P=1/2 E(X) = 1/2 Var(X) = 1/4 Binomial random variables Consider that n independent Bernoulli trials are performed Each of these trials has probability p of success and probability (1p) of failure Let X = number of successes in the n trialsG a y E n g in e e r s, In c T B P E R e g is tr a ti o n N o F1 0 4

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>0;x 0 E(X) = and ˙2 = 2 A brief note on the gamma function The quantity ( ) is known as the gamma function and it is equal to ( x) = Z 1 0 x 1e dx Useful result (1 2) = p ˇ If we set = 1 and = 1 we get f(x) = e x We see that the exponential distribution isP(X,Z) P(Z) and if the complete collection of all the RVs our agent is interested in is {X,Y,Z} then both the numerator and the denominator can be computed by marginalising the joint distribution P(X,Y,Z) In fact as the denominator serves essentially just toSafety How works Test new features Press Copyright Contact us Creators

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CT 85 °C, F 10 A T E X 0 5 8 X E C x F M G100 2X Ex db e b T5 T 6, t b I C T 1 0 °3 74 1 ' n 3 3 4 31 9 w 9 5 °L´evy’s martingale characterization of Brownian motion Suppose {Xt0≤ t ≤ 1} a martingale with continuous sample paths and X 0 = 0 Suppose also that X2 t −t is a martingale Then X is a Brownian motion Heuristics I’ll give a rough proof for why X 1 is N(0,1) distributed Let f (x,t) be a smooth function of two arguments, x ∈ R and t ∈ 0,1Define

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3 2;cos2 ax (75) Z cosaxdx= 1 a sinax (76) Z cos2T u2x) and the momentum density as p= u tu x (a) Show that @e=@t= @p=@xand @p=@t= @e=@x (b) Show that both e(x;t) and p(x;t) also satisfy the wave equation 3 Show that the wave equation has the following invariance properties (a) Any translate u(x y;t), where yis xed, is also a solution (b) Any derivative, say u x, of a solution is alsoX p (t) A 1 sin t A 2 cos t can also be written as x p (t) X cos t To convert between the 2 forms, ie to get the constants X and , substitute t=0 and equate the displacements and velocities in both equations This yields A X A X 21 cos , sin Thus 22 1 12 2, tan A X A A A

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