1. Consider two probability density functions on [0,1]: f0(x) = 3x2 and f1(x) = 4x3.
a.) Construct the most powerful test for H0 : X ~ f0 against HA : X ~ f1 with the significance level alpha = 0.1
b.) Find its power.
1. Consider two probability density functions on [0,1]: f0(x) = 3x2 and f1(x) = 4x3. a.)...
Problem 2: Consider two probability density function on [0,1] : fo(x) = 1 and fi(x) = 2x (a) Construct the most powerful test for Ho : X~ fo against H:Xfi with the signifi- cance level x = 0.1 (b) Find its power. (c) Suppose we observe X= 5/6. What is the p-value of the test in 2(a).
7. You have one observation Y , which has one of the discrete pdf’s y f0(y) f1(y) 0 0.1 0.3 1 0.1 0.1 2 0.1 0.1 3 0.1 0.2 4 0.2 0.1 5 0.1 0.1 6 0.3 0.1 You want to test H0 : f0 is true Ha : f1 is true (a) Here is a test: reject H0 if Y = 0, 1, 2, 3, or 5. What are the probabilities of the two types of error for this...
5. Suppose X a single observation from a population with a Beta(0,1) distribution. (a) Suppose we want to test Ho :0 <1 against H :0>1 an we use a rejection region of X > 1/2. Find the size and power function for this test. Sketch the power function. (b) Now suppose we want to test H, :0 = 1 against H :0 = 2. Find the most powerful level a test. (Is there a Theorem we can use?) (C) Is...
Let X be a random number from (0,1). Find the probability density functions of the random variables
1. Consider the following two probability density functions: f(3) = 2053 } for a <I<02 and g() = where ci and ca are finite real numbers. 265. for <y<02, (a) Show that f(r)dx = 9(r)dt = 1. (b) Find the cumulative distribution functions F(x) and Gu). (d) Show that if X-f(x), then 1-X g(x). (e) Show that if X h(x) = 21, for 0 <<1, then Y = c +(2-c)X ~f. (h) Show that if Uſ and U2 are two...
Problem 5 (15pts). Suppose that we observe a random sample X. from the density Xn 1 0 2 0, else, where m is a known constant which is greater than zero, and 0>0. (a) Find the most powerful test for testing Ho : θ Bo against b) Indicate how you would find the power of the most powerful test when θ-e-Do not perform (c) Is the resulting test uniformly most powerful for testing Ho :0-00 against Ha :e> et Explain...
Let X ~ U[0,1] be a standard uniform random variable. Find the probability density functions (pdf's) of the following random variables: iii) Y = 1/x0.5
5. Suppose Y represents a single observation from the probability density function given by: Soyo-1, 0, 0<y<1 elsewhere Find the most powerful test with significance level a=0.05 to test: HO: 0=1 vs. Ha: 0=2.
5. Let X1, ..., X 100 be i.i.d. random variables with the probability distribution function f(x;0) = 0(1 - 0)", r=0,1,2..., 0<o<1 Construct the uniformly most powerful test for H, :0= 1/2 vs HA: 0 <1/2 at the significance level a =0.01. Which theorems are you using? Hint: EX = 1, VarX = 10.
Consider the following joint probability density function of the random variables X and Y : (a) Find its marginal density functions (b) Are X and Y independent? (c) Find the condition density functions . (d) Evaluate P(0<X<2|Y=1)