The expectation value of a function f(x), denoted by (f(x)), is given by +∞0 (f(x)) = f(x)\(x)|²dx Yn(x) = where (x) is the normalised wave function. A one-dimensional box is on the x-axis in the region of 0 ≤ x ≤ L. The normalised wave functions for a particle in the box are given by 88 √2/² Calculate (x) and (x²) for a particle in the nth state. sin ηπχ L n = 1, 2, 3, ....

Q: 1. (a) Let X be a continuous, non-negative (that is, P(X ≥ 0) = 1) random variable with PDF f. Show…

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Q: a random variable whose pdf is the form of f(x) = Cxm(1-x) where x E (0, 1) and m is a non-negative…

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Q: Theorem 2. If X has the Beta distribution as given by Equation (12), then E[X] = a +B (13) %3D Proof…

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Q: Let u(æ) = sin(x) and v() = a = 18 and f(x) (*)a u (2) -| v (2) = u'v – uv f'

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Q: Check that the standard normal PDF p(x)=e-r²/v2™ satisfies the equation I xp(x)dx = p(y). y>0. By…

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Q: Į c(x + y?) for 0 <x< l and 0 < y < 1, S(x, y) = otherwise. Determine (a) the conditional p.d.f. of…

A: The required pdf is given by: fy=∫01fx,ydx=∫01cx+y2dx=c∫01xdx+∫01y2dx=cx22+xy201=c12+y2

Q: • For random variables X and Y and scalar constants a and b, we have E[aX + bY] = aE[X] + bE[Y], ||…

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Q: Now fx,(x) = *(0,)(x), and Fx₁(x) =(1-e1⁰0*)(0, ∞)(x);

A: Given,f(x)=1100e-1100x         ; 0&lt;X&lt;∞

Q: 14) The PDF of a random variable X is expressed by the quadratic function fx(x) = ax(1-x); for 0<x<…

A: f(x)=ax(1-x); 0&lt; x&lt; 1

Q: 1ΑΔΒ = |1, - 1g|

A: For two sets A and B, their symmetric difference is defined as         A Δ B = (A – B) ∪ (B – A)

Q: Maximise the function f(x, y) = 4xy² subject to r + y? = 2.

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Q: et X be a random variable that follows the Beta distribution with parameters a > 1 and > 1. The pdf…

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Q: Let f(2) := 25. Show that f is invertible in a neighbourhood of i and calculate (f-1Y(/(i)).

A: Let f(z) = z5, show that f is invertible in a neighbourhood of i and calculate(f-1)' f(i)

Q: Let f(z) := z5 . Show that f is invertible in a neighbourhood of i and calculate (f-1)'(f(i)).

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Q: The joint pdf of Y, and Y, is given by (a) ,with support (b) , and 0 elsewhere. (a) 1/y2; (b) 0 < y…

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Q: Let X1,..., Xn be iid pdf Ka,c x-a-1, x >c, f (x) = 10, а,с x < c. and unknown parameters a and c.…

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Q: Suppose X₁ and X₂ have the joint par f(x1, x₂) fw (w) For constants w₁ > 0 and w₂ > 0, let W = W₁X₁…

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Q: 3. The Cov(X) of a random vector X'= (X₁, ..., Xn) is defined as E{(X-E(X))'(X-E(X)}. True or false,…

A: We know that, X' is a vector and Covariance of a vector X is equal to the variance of X.  Which…

Q: (1,1) S[3x*+5y+i(x°-y°)] dz along y=x Evaluate (0,0)

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Q: Let Wn denote the symmetric random walk starting at 0, that is p = 1/2 and Wo = 0. Let X denote the…

A: Note:-Since you have posted a question with multiple sub-parts, we will solve first three sub-parts…

Q: Assume that X and Y have joint pdf x + 3x?y if 0 }). 1. Simplify the final expression as much as…

A: Thus, P(X≤1/2 and Y ≥1/2) =3/8 =0.375

Q: 1. Evaluate the given integral Q. 2y Q -ẢI (3=rs) 0 Your answer (3zx - y) dz dx dy

A: Since you have asked multiple questions here, we will solve the first question for you. To get the…

Q: A random variable X has a cdf given by 0, x ) (c) Find P(X <3) ^^

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Q: x)=2x IT X Is betwe 0 T; T(X)=C wn valuation, while the seller does r e seller sets a price, and…

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Q: say f(3)=5, f'(3)=-2 and let g(x) =xf(x/2) find the linearization of g(x) at x=6

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Q: Random vector X has PDF fX(x) = {ca'x when 0≤x≤1 and 0 otherwise} where a=[a1,...,an]' is a vector…

A: To find the constant c in the given probability density function for a random vector X, we must…

Q: 6. If f(x,y) = 2 x>0; y>0; x+y<1 Show the conditional pdf of Y given x is uniform.

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Q: are defined c chy's mean

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Q: A signal x(t) u(t) e¯ «t is applied to a network having an impulse response (1) W u(t) e Wt. lere a…

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