Suppose the andom variable X has pmf 3 , n 0, 1,2 Find the median and...
2. (Discrete uniform). Consider the PMF P(X x)= for x 1,2,...0 _ You have a random sample of size three from this distribution: {2,3,10}. a. Find the method of moments estimate for 0 HINT: a very useful fact is that k1 n(n+1) 2 b. Find the MLE for 0 c. Which estimator is unbiased? d. Which estimator is preferred?
2. (Discrete uniform). Consider the PMF P(X x)= for x 1,2,...0 _ You have a random sample of size three from...
Suppose a random variable X has a pmf p(x) = [3^(x-1)] / [4^x] , x = 1, 2, ... (a) Find the moment generating function of X. (b) Give a realistic example of an experiment that this random variable can be defined from its sample space. (c) Find the mean and variance of X.
Consider a random variable X with PMF Px(X) ={ cx x = 1,2,...,5 I found the value of c to be 1/15 1. find the complete expression for the CDF of X. 2. Find E[X], the expected value of X.
Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we see the numerical value of ?, we then draw a random variable ? whose (conditional) PMF is uniform on the set {1,2,…,2?}. 1.1 Write down an expression for the joint PMF ??,?(?,?). For ?=1,2,… and ?=1,2,…,2?: ??,?(?,?)=? 1.2 Find the marginal PMF ??(?) as a function of ?. For simplicity, provide the answer only for the case when ? is an even number. (The...
The joint pmf of X and Y is defined by f(x,y)=,
x=1,2; y=1,2
(a) Find Cov(X,Y).
(b)Find E(X|Y=1)
x + 2y 18
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 < X < 3). (b) Find P(X < 1). (e) Find t such that P(X <t) = (d) After the value of X has been observed, let y be the integer closest to X. Find the PMF of the random variable y U (2) Suppose for constants n E R and c > 0, we have the function cr" ifa > 1 0, otherwise (a)...
Suppose that X follows a discrete random variable with the following pmf table. x 3 4 5 p(x) 0.3 0.4 0.3 Find the standard deviation of X. Use two decimal points.
Suppose that X and Y have joint pmf px yx,y) fxy-/39 for x 1,2 and y 2,3 0elsewhere). a) Determine the marginal pmfs px(x) and py(y) b) Determine the conditional pmf of px(xly). c) Are X and Y independent? Give a clear determination using probability formulas.
5. Let X be a discrete random variable with the following PMF: for x = 0 Px(x)- for 1 for x = 2 0 otherwise a) Find Rx, the range of the random variable X. b) Find P(X21.5). c) Find P(0<X<2). d) Find P(X-0IX<2)
5. (6 pts.) Suppose the random variable X has probability density function ()if 0, ifx<0. Find the mean and variance of X. Write the equation one needs to solve to find the median m of X. Do not solve the equation, but at least simplify it enough so that it does not involve any integrals.