Consider X-B(3,1/4). The p.d.f of this discrete r.v. X is:
Let ,
The p. d. f. of this discrete r. v. X is Binomial distribution with parameter n=3 and p=1/4
The p. d. f. of X is ,
; x=0,1,2,.....,n
= 0 ; otherwise
2.4 Consider the independent r.v.'s x,..,.X with the Weibull p.d.f. (i) Show that- is the MLE of θ.
3.18 Let the r.v. X has the Geometric p.d.f. (i) Show that X is both sufficient and complete. U(X )-1 (ii) Show that the estimate U defined by: estimate of 6 if X-1, and U(X) -0 if X 2 2, is an unbiased (iii) Conclude that U is the UNU estimate of θ and also an entirely unreasonable estimate.
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected lifetime. On the basis of the random sample Xi,..X from this distribution, derive the MP test for testing the hypothesis Ho:to against the alternative HA:-(>o) at level of significance o 3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected...
Problem 1 (11 pts] The independent r.v.'s X and Y have p.d.f. f(t) = et, t>0. Compute the probability: P(X+Y > 2). Hint: Use independence of X and Y in order to find their joint p.d.f., fx,y, and then use the diagram below to compute the probability: P(X+Y < 2). y 2 r+y = 2 y . ! 2 0 2-y Note: If X and Y represent the lifetimes of 2 identical equipment of expected lifetime 1 time unit, then...
2.3 Let X be a r.v. describing the lifetime of a certain equipment, and suppose that the p.d.f. of X is f (ii) We know (see Exercise 2.1) that the MLE of θ, based on a random sample of size n from the above pd.f., is θ = 1/ X. Then determine the MLE of g(9).
4. Let X have p.d.f. fx(1),-1 < 2. Find the p.d.f. of Y-X2
If f(x) and g(x) are p.d.f. on the same interval a ≤ x ≤ b, show that f(x)+g(x) is NOT a p.d.f. on [a,b]. In addition, show that the function λf(x)+(1−λ)g(x) is a p.d.f. on [a,b] for any λ ∈ [0,1].
A Consider the vector x=(3,1) What is the vectorial representation of x wing using Basis B= {(1;-), CUM 2) What is the vectorial representation of x cuing Basis Ba={(1,1),(0,2}} 3) Build a linear mapping that goes from Basis By to the Basis Br. ( Find the matrix P x asociated with the mapping) u Verify that px, where x is the vectorial reprental.com of part 1
(Discrete R.V. and Probability Mass Function) Suppose you work at a banana factory, where there is a conveyor belt along which bananas are always coming your way. Each banana is rotten w.p. 0.3, independent of all else. The first time that you get a rotten banana, you have to shut down the plant operations for the day. Until that time, you process the fresh bananas you receive as follows. The first 90 fresh bananas you process must be packed into...
Let X be a r.v. with probability density function f(x)-e(4-x2), -2 < otherwise (a) What is the value of c? (b) What is the cumulative distribution function of X? (c) What is EX) and VarX