1. Consider a Bernoulli random variable X with parameter π. Show that Var[X] =π(1 − π).
1. Consider a Bernoulli random variable X with parameter π. Show that Var[X] =π(1 − π).
Recall that a Bernoulli random variable with parameter p is a random variable that takes the value 1 with probability p, and the value 0 with probability 1 - p. Let X be a Bernoulli random variable with parameter 0.7. Compute the expectation values of X, denoted by E[X*1, for the following three values of k: k = 1,4, and 3203. E [X] = E [X4 E [X3203
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if X {0,1} and P(X-1)-θ. Suppose that we have nd random variables y, x, following a Bernoulli(0) distribution and observed values y1,... . Jn a) Show that EIX) θ and Var[X] θ(1-0). b) Let θ = ỹ = (yit . .-+ yn)/n. Show that θ is unbiased for θ and compute its variance. c) Let θ-(yit . . . +yn + 1)/(n + 2) (this...
Consider the random sum S= Xj, where the X, are IID Bernoulli random variables with parameter p and N is a Poisson random variable with parameter 1. N is independent of the X; values. a. Calculate the MGF of S. b. Show S is Poisson with parameter Ap. Here is one interpretation of this result: If the number of people with a certain disease is Poisson with parameter 1 and each person tests positive for the disease with probability p,...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
6. Let X be an exponential random variable with parameter 1 = 2. Compute E[ex]. = 7. Consider a random variable X with E[X] u and Var(X) 02. Let Y = X-4. Find E[Y] and Var(Y). The answer should not depend on whether X is a discrete or continuous random variable.
X is a Poisson random variable of parameter 3 and Y an exponential random variable of parameter 3. Suppose X and Y are independent. Then A Var(2X + 9Y + 1) = 22 B Var(2X + 9Y + 1) = 7 CE[2X2 + 9Y2] = 19 D E[2X2 + 9Y2] = 26
Problem 4: Consider the problem of estimating the unknown parameter p of a Bernoulli random variable that describes the probability that a coin toss results in a head. Denote by X the outcome of the jth toss of the coin and let j-1 denote the sample mean. Part I: Use Chebyshev inequality to determine the number of tosses n needed so that P( -pl> 0.01) 0.01 The estimate should be independent of p Part II: Compute ElIX -pl]. Your answer...
The random variable Q is uniform on [0,1] . Conditioned on Q=q , the random variable X is Bernoulli with parameter q . A uniform random variable on [0,1] has a variance of 1/12 and also satisfies E[Q2]=1/3 . Then: Var(E[X|Q])= ? E[Var(X|Q)]= ?
Consider a random variable X with the following properties E[X] - 10 and var(X) - 9. Consider a new random variable such that Y-1-5X Calculate the following (a) EY] - (b) var(Y) = 5
7 Let X be a Bernoulli random variable with P(X = 1) = 4. Consider a sample from X of size n. Formulate both the LLN and the CLT. Remember that you must calculate the mean and the variance of X to answer this question