If X and Y are independent exponential random variables, each having parameter λ. (a) Find the joint density function of U = X + Y by using the convolution of fX and fY . (b) Find the joint density function of V = X − Y by using the method of transformation
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If X and Y are independent exponential random variables, each having parameter λ.
(a) Find the joint density
(b) Find the joint density function of V = X − Y by using the method of transformation.
(c) Are U and V independent?
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- Concentration of a solute According to the Ficks law, the diffusion of a solute across a cell membrane is given by c(t)=kAV[Cc(t)],(1) Where A is the area of the cell membrane, V is the volume of the cell, ct is the concentration inside the cell at time t, C is the concentration outside the cell, and k is constant. If co represents the concentration of the solute inside the cell when t = 0, then it can be shown that c(t)=(c0C)ekAt/V+C.(2) a. Use the last result to find c(t). b. Substitute back into Equation 1 to show that 2 is indeed the correct antiderivative of 1.Answer all three questions in this section 2. Let V and W be two random variables with joint probability density function given by fvw (v,w) = aw exp(-dv +(8-A)w²), w>0, v> w², where A, & are positive constants. (a) Find the value of a. (b) Consider the transformation X = WP, Y=V_W². Find the joint density of X and Y. Hence show that X and Y are independent exponential random variables. (c) State the distribution of AX + SY.Let X and Y be independent normally distributed random variables with mean zero and variances og = 1 and of = 4. (a) Write the joint probability density function fx.y (r, y). • (b) Define new random variables U = aX + Y and V = X – Y, where a + -1 is a real number. Find the absolute value of the Jacobian of transform from X, Y to U, V. (c) Find the joint probability density function for U and V. Find a for which U and V are independent random variables. Write down fu,v (u, v) for this a in the answer.
- i) Find the joint probability density function of X1 and X2. ii) Using the transformation method, find the joint probability density function of Y1 = X21 and Y2 = X1 + X2 iii) Determine the marginal density function of Y21)A random vector z = (x y) T has joint probability density given by:a)Sketch the graph of the probability density function pxy(X, Y ).b)Determine the value of the constant C.c)Determine the mean mz vector.a) Find the joint density function of X and Y. b) Using the method of transformation, find the joint density function of random variables U and V where U = X^2 Y and V = Y .
- Let X and Y be two independent random variables that follow exponential distribution Exp(0), with parameter 0>0. Let U = X+Yand V = X-Y. • [(a)] Find the Jacobian of the transformation ((b)] Determine the joint pdf, g(u, v), of the bivariate vector (U,V). [(c)] Find the marginal pdfs gu (u) and gy (v). [(d)] Are U and V independent? Justify your answer. SOLVE = %3D CALC Paragraph BIEE Fir (- STO RC CONS 4Let X and Y be two independent random variables that follow exponential distribution Exp(0), with parameter 0 > 0. Let U = X+Y and V = X- Y. [(a)] Find the Jacobian of the transformation [(b)] Determine the joint pdf, g(u, v), of the bivariate vector (U,V). [(c)] Find the marginal pdfs gu (u) and gy (v). [(d)] Are U and V independent? Justify your answer.c) Let X [0,1]x[0,1]x[0,1]. Let Y = g(X), be a two dimensional random vector, where 91(X ) = X1 + X2 , g2(X) = X2 – X3 . Find the density function of Y. [X1,X2, X3] be a three dimensional random vector uniformly distributed on
- Let X ∼ Unif(0, 1). Let g(x) = e x and Y = g(X). (i) Find the density fX of the random variable X. (ii) Find the cdf FY (y) = P(Y ≤ y) of Y . (iii) Find the density fY of the random variable Y .Exercise 40 Let X Unif(0,1). Let g(x) = e" and Y = g(X). (i) Find the density fx of the random variable X. (ii) Find the cdf Fy (y) = P(Y < y) of Y. (iii) Find the density fy of the random variable Y.(11) Let X, Y be independent random variables with X ~ Exp(A1) and Y - Erp(A2). Find the joint density of Z = X +Y,W = X – Y. The following answers are provided. Read the answers carefully before making your choice. (a) The inverse transformation is I = v (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2. The jacobian of the transformation is the following determinant J = = The bivariate density of Z, W is g(z, w) = A1A2e 1(=+w)/2e¬A2(z¬w)/2 |J], if 쁘 > 0, > 0. This is equivalent to saying that if – z 0, " > 0. This is equivalent to saying that if – z 0, 를 >0. This is equivalent to saying that if - 00 0," > 0. This is equivalent to saying that if – z 0, > 0. This is equivalent to saying that if – z< w < z, 0 < z g(z, w) = if otherwise, (a) (b) (c) (d) (e) N/A (vii- Select One)
