2. The continuous random variables X and Y have known joint probability density function f(x,y) given by fw(x, y) = {(x+y)/8 0≤x≤ 2,0 ≤ys 2 0 otherwise. Define the random variable z as Z = min(X,Y). a) Determine the probability density function (²) 7 f₂(z) of the random variable Z. b) Determine the probability
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- Let X1 and X2 be two continuous random variableshaving the joint probability density f(x1, x2) = 4x1x2 for 0 < x1 < 1, 0 < x2 < 10 elsewhereFind the joint probability density of Y1 = X21 and Y2 = X1X2.The random variable Y has probability density function f(V) = k(y + y³), 0 2. Hence find PG < Y <). iii) Find the variance of Y.The life lengths of two transistors in an electronic circuit is a random vector (X; Y ) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y ) is given by ´2e-(x+2) x 2 0, y 20 fx,y(x,y)=| else Then the probability that the first transistor burned during half hour given that the second one lasts at least half hour equals Select one: a. 0.3935 b. 0.606 c. 0.7772 d. 0.3669 e. 0.6318
- The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.The life lengths of two transistors in an electronic circuit is a random vector (X; Y ) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y ) is given by | 2e-(x+2y) x 2 0, y 20 fx,y(x,y) = fx.MX.v) else Then the probability that the first transistor last for at least half hour given that the second one lasts at least half hour equals Select one: a. 0.3669 b. 0.3935 c. 0.7772 d. 0.6318 e. 0.606Let Xbe a continuous random variable with density f (x) = 24x-4 for x > 2. Then Var (X) is equal to
- Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Var(a + X) = Var (X)Let X and Y be two random variables that are independent and have the same probability density function. If U = max (X, Y) V = min (X, Y), find the distribution function of the U random variable using the distribution function technique.The life lengths of two transistors in an electronic circuit is a random vector (X; Y) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y) is given by x 2 0, y 2 0 fx.,fx.v) = 20 else Then the probability that the first transistor burned during half hour given that the second one lasts at least half hour equals Select one: a. 0.606 b. 0.3935 C. 0.6318 d. 0.3669 e. 0.7772
- Let X be a random variable with density function - 1< x < 2, fn) = fx*/3, f(x) = 0, elsewhere. Find the expected value and the covariance of g(X) = 4X + 3.10) Let f(x) by the probability density function of some continuous random variable. Explain why it must be true that f(x) dx =1.Determine the conditional probability distribution of Y given that X = 2. Where the joint probability density function is given by f(x,y)= 1 - (x + y) for 1 < x < 4 and 0 < y < 3.