A random variable Y has the density function f(y) = 1/3e^(y/3)
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A random variable Y has the density function f(y) = 1/3e^(y/3) Thank you! 1. (15 points points) A random variable Y has...
1. Let X be a continuous random variable with probability density function f(x) = { if x > 2 otherwise 0 Check that f(-x) is indeed a probability density function. Find P(X > 5) and E[X]. 2. Let X be a continuous random variable with probability density function f(x) = = { SE otherwise where c is a constant. Find c, and E[X].
5. (28 points) A continuous random variable X has probability density function given by f(x) = 3x^2,0<x< 1 O otherwise (c) What is the c.d.f. of Y = X^2 - 1? What is the p.d.f. of Y = X^2 - 1?
The density of random variable X is f(x) = 5(Xº+1)(3-X) for 1<x<3 and 0 otherwise. Using the R integrate function: 68 a) Find the probability that X > 2.10 b) Find the probability that 1.5 < X < 2.5 c) Find the expected value of x d) Find the standard deviation of X e) In the following paste your R script for this problem
Problem 2: 20 points 10 5 + 5) A continuous random variable (Y) has a density, fY (3e-3V for y>0 and f () 0, elsewhere. Given Y y, a discrete random variable, N, is Poisson distributed with the rate equal to y TA 1. Derive the marginal distribution of N 2. Determine the marginal expectation of N, EIN 3. Determine the marginal variance of N, Var[N]
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
Suppose that a random variable X has a (probability) density function given by 52e-2, for x > 0; f(x) = 0, otherwise, (i) Calculate the moment generating function of X. [6 marks] (ii) Calculate E(X) and E(X²). [6 marks] (iii) Calculate E(ex/2), E(ex) and E(C3x), if they exist. [3 marks] (iv) Based on an independent random sample X = {X1, X2, ..., Xn} from the dis- tribution of X, provide a consistent estimator for 0 = E(esin(\)), where sin() is...
Let X be a continuous random variable with the following density function. Find E(X) and var(X). 6e -7x for x>0 f(x) = { for xso 6 E(X) = 49 var(X) =
The random variable X has the probability density function (x)a +br20 otherwise If E(X) 0.6, find (a) P(X <름) (b) Var(x)
3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in...