Suppose S= [0,1) with the uniform probability measure and X(s) =s. Calculate MX(z).
Suppose S= [0,1) with the uniform probability measure and X(s) =s. Calculate MX(z).
Exercise 5.13. Suppose S = [0, 1) with the uniform probability measure and X(S) s. Calculate Mx(2).
The numbers x, y, z are independent with uniform distribution on [0,1]. Find the probability that one can construct a triangle with sides length x, y, z.
Let X, Y, Z be independent uniform random variables on [0,1]. What is the probability that Y lies between X and Z.
Exercise 5.12. Let n∈N and S={0,1,...,n−1}, and suppose that P({s}) = 1/n for each s∈S. Let X:S→R be the random variable defined by X(s) =s. i) Find a closed form formula for MX(z), which does not use sigma notation or any other iterative notation. (ii) Calculate E[X]. (iii) Calculate Var[X].
Problem 3 Let X be Uniform(0,1) and Y be Exponential (1). Assume that X and Y are independent. i. Find the PDF of Z- X +Y using convolution. ii. Find the moment generating function, øz(s), of Z. Assume that s< 0. iii. Check that the moment generating function of Z is the product of the moment gen erating functions of X and Y Problem 3 Let X be Uniform(0,1) and Y be Exponential (1). Assume that X and Y are...
Problem 2 Suppose X ~Uniform[0,1 (1) What is the density function? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(x)-P(X x) for x E [0, 1]. (4) Let Ylog X. Calculate F(-P(Y 3 y) for y 20. Calculate the density of Y.
4. Let Z ~ N(0,1) be a standard normal variable. Calculate the probability (a) P(1 <Z < 2). (b) P(-0.25 < < < 0.8). (c) P(Z = 0). (d) P(Z > -1).
Let X and Y be iid uniform random variables on [0,1]. Find the pdf of Z=X+Y
If the two probability variables X and Y follow U (0,1) independently of each other, calculate the probability density function of Z = X - Y.
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b) Obtain the marginal pdf of S. 6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b)...