assume that s and t are independent exponentially distributed, set u = min(s,t), v = max(s,t)....

assume that s and t are independent exponentially distributed, set u = min(s,t), v = max(s,t). prove that u and w = v-u are independent

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orchestra answered 1 year ago

Calculus dictates that (∂U/∂V) T,Ni = T(∂S/∂V)T,Ni – p = T(∂p/∂T)V,Ni – p (a) Calculate (∂U/∂V)...

Calculus dictates that (∂U/∂V) T,Ni = T(∂S/∂V)T,Ni – p = T(∂p/∂T)V,Ni – p (a) Calculate (∂U/∂V) T,N for an ideal gas [ for which p = nRT/V ] (b) Calculate (∂U/∂V) T,N for a van der Waals gas [ for which p = nRT/(V–nb) – a (n/V)2 ] (c) Give a physical explanation for the difference between the two. (Note: Since the mole number n is just the particle number N divided by Avogadro’s number, holding one constant is equivalent...

Let U, V be iid Unif(0, 1) random variables, and set M = max(U,V) and N...

Let U, V be iid Unif(0, 1) random variables, and set M = max(U,V) and N = min (U,V) (a) Find the conditional density of N given M = a for any value of a ∈ (0, 1). (b) Find Cov(M, N).

(1) Suppose that V is a vector space and that S = {u,v} is a set...

(1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates V...

Let Zt = U sin(2pit) + V cos(2pit), where U and V are independent random variables,...

Let Zt = U sin(2*pi*t) + V cos(2*pi*t), where U and V are independent random variables, each with mean 0 and variance 1. (a) Is Zt strictly stationary? (b) Is Zt weakly stationary?

The differential for the Internal energy, U, at constant composition is ?U = −P?V + T?S...

The differential for the Internal energy, U, at constant composition is ?U = −P?V + T?S (a) What are the natural independent variables of U? [2] (b) Derive an expression for the change in internal energy at constant Volume starting with the above differential for the internal function, U, at constant composition. [3] (c) Using the criterion for exact differentials, write the Maxwell relation that is derived from this equation. [2] (d) Based on your answer in part (a), write...

1. Let U = {r, s, t, u, v, w, x, y, z}, D = {s,...

1. Let U = {r, s, t, u, v, w, x, y, z}, D = {s, t, u, v, w}, E = {v, w, x}, and F = {t, u}. Use roster notation to list the elements of D ∩ E. a. {v, w} b. {r, s, t, u, v, w, x, y, z} c. {s, t, u} d. {s, t, u, v, w, x, y, z} 2. Let U = {r, s, t, u, v, w, x, y, z},...

Let U and V be independent continuous random variables uniformly distributed from 0 to 1. Let...

Let U and V be independent continuous random variables uniformly distributed from 0 to 1. Let X = max(U, V). What is Cov(X, U)?

3.5.4 ([Ber14, Ex. 3.6.14]). Let T : V → W and S : W → U...

3.5.4 ([Ber14, Ex. 3.6.14]). Let T : V → W and S : W → U be linear maps, with V finite dimensional. (a) If S is injective, then Ker ST = Ker T and rank(ST) = rank(T). (b) If T is surjective, then Im ST = Im S and null(ST) − null(S) = dim V − dim W

Let U and V be vector spaces, and let L(V,U) be the set of all linear...

Let U and V be vector spaces, and let L(V,U) be the set of all linear transformations from V to U. Let T_1 and T_2 be in L(V,U),v be in V, and x a real number. Define vector addition in L(V,U) by (T_1+T_2)(v)=T_1(v)+T_2(v) , and define scalar multiplication of linear maps as (xT)(v)=xT(v). Show that under these operations, L(V,U) is a vector space.

Lifetime of certain device is exponentially distributed random variable T( λ ). Probability that T >...

Lifetime of certain device is exponentially distributed random variable T( λ ). Probability that T > 10 is e-3: P{ T> 10} = e-3 The system consists of 6 devices of such type and is in working condition if all of its components are in working condition. a) Find expectation μx and standard deviation σx for given distribution. b) Find the probability that the system will not fail I) in the next 3 years? II) in the next 8 years?...

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