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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.Let x be a continuous random variable with the density function: f(x) = 3e-3x when x>0 and 0 else Find the variance of the random variable x.• Find the density of Z = (X+ Y)2, where X and Y are independent uniform random variables over (-1, +1).
- Let Xbe a continuous random variable with density f (x) = 24x-4 for x > 2. Then Var (X) is equal toSuppose that X is a normal random variable with parameters u = 5 and o = 1. Define Y = X2 – 10X + 25. Then what is the probability density function of Y (fy(y)). Probability TheoryLet 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.
- Be X a random variable with density function defined by f(x) = -2², , x>0. 3 Get the moment-generating function and based on it, calculate the average and variance of X.Let X and Y be independent random variables with density f (x) = 3x² for 0 < x < 1. Then P (X+ Y < 1) is equal tofunction given by Suppose that X and Y have a joint probability density 7e-2-7y if x, y ≥ 0 otherwise fxy(x, y) = = (d) Are the random variables X and Y independent? Justify your answer.
- 1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.The density function of two random variables X and Y is fxy (x, y) = 16 e-4 (x+y). u (x) u (y) Find the mean of the function (X, ¥)= 5 for 0 < X s; 1 = -1 for < X and lor< Y 2 = 0 for all other X and Y
