Suppose X1, X2, Xz~exp(1) and they are independent. (a) Compute the cdf of X1 (b) Let...
4. Suppose Xi, X2, X3 ~exp(1) and they are independent (a) Compute the edf of X (b) Let Y max(Xi, X2, X3). Find the cdf of Y. (c) Derive the pdf of Y
2. Let Xi exp(1) and X2 ~ variables with rate 1. Let: erp(1) be independent and identically-distributed exponential random (a) What is the cdf of X1? b) What is the joint pdf of (Xi, X2)? (c) What is the joint pdf of (Y, Z)? d) What is the marginal pdf of z?
Let X1 , X2 , and X3 be independent and uniformly distributed between -2 and 2. (a) Find the CDF and PDF of Y =X1 + 2X2 . (b) Find the CDF of Z = Y + X3 . (c) Find the joint PDF of Y and Z . (Hint: Try the trick in Problem 2(b))
Suppose X = Exp(1) and Y= -ln(x)
(a)Find the cumulative distribution function of Y .
(b) Find the probability density function of Y .
(c) Let X1, X2, ... , Xk be i.i.d. Exp(1), and let Mk =
max{X1,..... , Xk)(Maximum of X1, ..., Xk). Find the probability
density function of Mk.(Hint: P(min(X1, X2, X3) > k) = P(X1
>= k, X2 >= k, X3 >= kq, how about max ?)
(d) Show that as k → 00, the CDF...
6. (10 points) Suppose X – Exp(1) and Y = -In(X) (a) Find the cumulative distribution function of Y. (b) Find the probability density function of Y. (c) Let X1, X2,...,be i.i.d. Exp(1), and let Mk = max(X1,..., Xk) (Maximum of X1, ..., Xk). Find the probability density function of Mk (Hint: P(min(X1, X2, X3) > k) = P(X1 > k, X2 > k, X3 > k), how about max ?) (d) Show that as k- , the CDF of...
Let X1 d = R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of...
Let X1 , X, , and X3 be independent and uniformly distributed between-2 and 2. (a) Find the CDF and PDF ofYX, +2X2 (b) Find the CDF of Z-), + X, . (c) Find the joint PDF of Y and Z.(: Try the trick in Problem 2(b)
Let X1 , X, , and X3 be independent and uniformly distributed between-2 and 2. (a) Find the CDF and PDF ofYX, +2X2 (b) Find the CDF of Z-), + X, . (c)...
Let X1,X2 be two independent
exponential random variables with λ=1, compute the
P(X1+X2<t) using the joint density function. And let Z be gamma
random variable with parameters (2,1). Compute the probability that
P(Z < t). And what you can find by comparing P(X1+X2<t) and
P(Z < t)? And compare P(X1+X2+X3<t) Xi iid
(independent and identically distributed) ~Exp(1) and P(Z < t)
Z~Gamma(3,1) (You don’t have to compute)
(Hint: You can use the fact that Γ(2)=1,
Γ(3)=2)
Problem 2[10 points] Let...
Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of Y...
7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y = max(X,Xy) and Z-X1 + X2. (1) Calculate the joint PDF of Y and Z. (2) Derive the marginal PDF of both Y and Z. Are Y and Z independent?
7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y...