( ) F α. a. Sketch! r as a function of r for fixed θ. For the sketch, assume that θ is roughly the same ( )

Transcription

1 . An acoustic a eflecting off a wav bounda (such as the sea suface) will see onl that pat of the bounda inclined towad the a. Conside a a with inclination to the hoizontal θ (whee θ is necessail positive, and is taken to be deteministic), and suppose that the bounda inclination (which ma have an sign, and is taken to be andom) is nomall distibuted. Let be the bounda inclination seen b the a. Then the pobabilit densit function (pdf) of is p F α = D( θ ), θ,, > θ whee p is the nomal pdf and D is the nomal cumulative distibution function, each with zeo mean and standad deviation σ. Assume that σ π. a. Sketch as a function of fo fixed θ. Fo the sketch, assume that θ is oughl the same F α size as σ. b. The pdf includes a nomalization b. What is the pupose of this F α nomalization? c. What is the mode of? d. What is α the mean of? You solution can be witten in tems of p θ,, and σ. What is the sign of α? Does ou solution agee with ou sketch? e. What is the limit on θ in which the mean and mode of appoach equalit? D( θ ) D( θ )

2 a F α () θ b. The pupose of the nomalization is to make the integal of the pdf equal to one. c. The mode of is zeo. d. The mean of is α = F α d θ = p D θ = D θ d θ D( θ ) = σ p θ σ π exp σ d The mean of negative, in ageement with sketch. e. The mean and mode of appoach equalit in the limit θ σ.

3 . Conside two independent andom vaiables x and, unifoml distibuted between and a. What is the pobabilit densit function of the vaiable z = x? An appoach to finding the pobabilit densit function is fist to find the cumulative distibution function of z, and then integate. a s=t/ s t/a a The joint pdf of x and has the value a - and the integal is ove the shaded aea in the figue. t a a t D z ( t) = a ad + d whee the fist tem is the integal ove the ectangula egion fo t, and the second tem is a the integal unde the cuve s = t fo lage values of. Evaluating the integals ields D z ( t) = t log t a a Taking the deivative gives the answe we seek ( t) = a log t fo t a. F z t a a

4 3. Conside the function whee is a eal vaiable, and c is a constant. a. What must the value of c be fo h() to be a valid pobabilit densit function. b. Calculate the fist two moments of h(). c. Using a compute, geneate andom vaiables govened b h() and plot a histogam. Hint: use the and function in Matlab. d. B summing a numbe of these andom vaiables, show the appoach to a nomal distibution as suggested b the cental limit theoem. Plot the pdfs of sums of,, and andom vaiables. 3a. Fo h() to be a valid pdf, its integal must equal. Since c must be equal to /. = h c,, elsewhee hd = cd + cd = c 3b. The fist moment of h is The second moment of h is µ = hd = d + σ = hd = d + d = d = 7 3 3c. Suppose we can geneate unifoml distibuted andom numbes w in the ange to. This is accomplished in Matlab using the and function. Then we can geneate andom numbes x govened b the distibution h in two steps. Fist, geneate andom numbes z unifoml distibuted between / and /: z = w Then ceate x as x = z + sign(z) pdf(x) x

5 3d. Ceate a sum of the andom vaiables x using = ( Nσ ) x i The nomalization causes to have unit vaiance. N i=.4 Sum of andom vaiables.7 Sum of andom vaiables..6.5 pdf().8.6 pdf() Sum of andom vaiables.35.3 pdf()

6 4. Geneate joint-nomall-distibuted vaiables x and such that x = =, x =, =.5, and x =.5. Do this b geneating two independent nomall distibuted vaiables, then scaling and otating them. a. Make a scatte plot of, ealizations of x and. b. Calculate and plot the joint pdf fom these same, ealizations. c. B diect calculation fom these, ealizations pove that the means, vaiances, and covaiance of x and ae coect. Conside two independent, zeo-mean, nomall distibuted andom vaiables ˆx and ŷ. Define complex andom vaiables z and ẑ as follows z = x + i ẑ = ˆx + iŷ so that the desied otation can be witten z = ẑe iθ and ˆx + iŷ = ( x + i)e iθ = x cosθ + sinθ Then the covaiance of ˆx and ŷ is ˆxŷ = x ( cos θ sin θ) + ( x )cosθ sinθ Since ˆxŷ =, and using two tigonometic identities, we find the esult + i( cosθ xsinθ ) θ = x tan x Substituting the desied values fo the vaiances and covaiance, we find θ=.596. The vaiances of ˆx and ŷ ae ˆx = x cos θ + x cosθ sinθ + sin θ ŷ = cos θ x cosθ sinθ + x sin θ Plugging awa, we find ˆx / =.464 and ŷ / =.344, giving the appopiate scaling. Run the scipt sol_jointnomal.m to geneate the andom vaiables and make plots.

7 4a x 5 4b x 5

8 4c. Fo the, ealizations in the scatte plot, the vaiances and covaiances ae: <x>= -. <>= <x^>=.988 <^>=.49 <x>= Check.