5.2.4. Show that for any node distribution and any x e[to, t,), (5.2.8) k=0 int: The simplest way is to apply (6.2.5) This is called the partition of unity property. (5.2.5) 5.2.4. Show that for...
how that for any node distribution and any xeLo,tn, H()-1 (5.2.8) k-0 (Hint: The simplest way is to apply (5.2.5).) This is called the partition of unity propertv 72 (5.2.5) y is reflected in Function 5.2.2. Take note that the The resulting algorithmic simplicit output of Function 5.2.2 is itself a function, meant to be called with a single (possibly vector) argument representing values of x. Our mathematical viewpoint is that the result of an interpolation process is a function,...
"k)-T, E(X"k+1)-0, k = 0.1, m.g.f. of X and also its ch.f. Then deduce the distribution of X. 6. Let X be a r.v. such that E(X Find the
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show that the posterior distribution of 0 is Gamma(nTk, n ). (b) [4 marks Find the probability function of the marginal distribution of Y = nX. (Note that the conditional distribution of on Y is not the same X1, ..., Xn.) as on
iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show...
where
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
Problem 0.1 Let X be the number of people who enter a bank by time t>0. Suppose ke-t k! for k 0,1,2,., and for t>s > 0, and k-r=0,1,2, . . . . (a) Find Pr(X2 = k | X,-1) for k = 0, 1, 2, . . . . (b) Find E[X2 X1-1 Useful information: Don't eat yellow snow, and et-L=0 tk/k! Problem 0.2 Recall the Geometric(p) distribution where Xnumber of flips of a coin until you get a...
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Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Recall that the one-sided Laplace transform of x(t)is defined as x(s)-J x()e "atfor any complex numl 1-0 0C A special case of this is X(ia) x(t)e-ω'dt, which is called the one-sided Fourier transform (FT) of x( 1-0 transforms the time domain into the frequency domain; a domain often preferred by engineers, as it decom its various frequency components. Consider the following approximation of the unit impulse δφ : x(t)-[u, (t)-4(t-A)] / Δ , where Δ is the pulse width. (a)...
4. Assume f and |fl are both integrable on la, b], a < b, show f(x) dx sf( d. Hint: there are several approaches. One good way is to apply the triangle inequality of series TL T& a where a, E R, ail s on the Riemann sums.
4. Assume f and |fl are both integrable on la, b], a
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...