Hw3
.pdf
School
University of Illinois, Chicago**We aren't endorsed by this school
Course
STAT 401
Subject
Statistics
Date
Dec 29, 2024
Pages
2
Uploaded by phyllocactus511
Homework 3 of Stat 401, Spring 2024 (due Jan 31 in class) 1.4.8. In a certain factory, machines I, II, and III are all producing springs of the same length. Machines I, II, and III produce 1%. 4%, and 2% defective springs. respectively. Of the total production of springs in the factory, Machine I produces 30%. Machine II produces 25%. and Machine III produces 45%. (a) If one spring is selected at random from the total springs produced in a given day, determine the probability that it is defective. (b) Given that the selected spring is defective, find the conditional probability that it was produced by Machine II. 1.4.10. In an office there are two boxes of thumb drives: Box A; contains seven 100 GB drives and three 500 GB drives, and box A, contains two 100 GB drives and eight 500 GB drives. A person is handed a box at random with prior probabilities P(A;) = % and P(A,) = %, possibly due to the boxes’ respective locations. A drive is then selected at random and the event B occurs if it is a 500 GB drive. Using an equally likely assumption for each drive in the selected box, compute P(A;|B) and P(As|B). 1.4.25. Let the three mutually independent events C, Cs, and C3 be such that P(Cy) = P(Cy) = P(Cy) = i. Find P[(C{ N CS) U Cs]. 1.4.34. A chemist wishes to detect an impurity in a certain compound that she is making. There is a test that detects an impurity with probability 0.90; however, this test indicates that an impurity is there when it is not about 5% of the time. The chemist produces compounds with the impurity about 20% of the time. A compound is selected at random from the chemist’s output. The test indicates that an impurity is present. What is the conditional probability that the compound actually has the impurity? 1.5.3. Let px(z) = /15, = 1,2, 3,4, 5, zero elsewhere, be the pmf of X. Find P(X=1or2),P(3<X<2),and P(1<X <2).
1.5.6. Let the probability set function of the random variable X be Px (D) = [ f(x)dx, where f(z) = 22/9, for x € D = {x : 0 < & < 3}. Define the events Dy ={x:0<2x <1} and Dy = {2 :2 < 2 < 3}. Compute Px(D1), Px(Ds), and Px (D U D). 1.5.8. Suppose the random variable X has the cdf 0 r<—1 F(z) = “’TQ -1<z<1 1 1 <. Write an R function to sketch the graph of F'(x). Use your graph to obtain the probabilities: (a) P(—3 < X < 3); (b) P(X =0); (¢) P(X =1); (d) P(2 < X < 3).