Solutions to Homework #8
(W4150 Intro to Probability and Statistics, S04) Sec. 9.7. (4) n = 30; X = 780; = 40; z/2 = z0.02 = 2.054 Hence by applying formula for CI from p.235 we obtain: 40 40 ) or 765 < < 795 ) < < 780 + (2.05)( 780 (2.05)( 30 30
Sec.9.7.
(W4150 Intro to Probability and Statistics, S04) Sec. 4.1. (1) In this problem we have to integrate joint distribution density function multiplied by x with respect to both x and y - but one should be very careful when setting limits for integration. You
(W4150 Intro to Probability and Statistics, S04) Sec. 2.5. (8) Let A = cfw_there is a defect in braking system and B = cfw_there is a defect in fueling system. We are given that P(A) = .25, P(B) = .17 and P(A B) = .15. Then: (a) P(A B) = P(A) + P(B) - P(A
1
Expected Values of Random Variables
Often it is convenient to describe a random variable using some location measure. The most important location measure is the expected value (a.k.a. the mean or the weighted average). For a discrete random variable X i
1
Discrete Random Variables
We will now review a number of well known discrete random variables. In each case we will begin with an example. We will then give the denition, often through the specication of the probability mass function, and will compute t
1
Continuous Random Variables
In this lecture we introduce a number of continuous random variables that are useful in practice.
1.1
The Uniform Distribution
We say that U has a uniform distribution over the interval [0, 1] if each subinterval of the same
1
Functions of Random Variables
We have encountered functions of random variables before when we were interested in nding the expectation of Y = g (X ). Now, we are interested in nding the distribution of Y .
1.1
Discrete Case
P (Y = y ) = P (X cfw_x : g
1
Random Sampling
A population consists of the totality of the observations with which we are concerned. The size of the population can be nite or innite. Examples of nite populations are the height of the students at Columbia University, the lengths of s
1
Linear Regression
Suppose that (xi , yi ) i = 1, . . . , n denote the height in inches of adult male students and their fathers. If we plot (xi , yi ) i = 1, . . . , n we would see a linear pattern with taller fathers having taller sons. This linear pat
1
Random Variables
In applications we are interested in quantitative properties of experimental results. Example: Toss a coin three times and count the number of heads. The sample space is S = cfw_(t, t, t), (t, t, h), (t, h, t), (h, t, t), (t, h, h), (h,
1
Lecture Plan
Finite Sample Space (Review). Conditional Probability Independence Random Variables Expected Values Markovs Inequality
2
Finite Sample Space (Review)
S = cfw_w1 , w2 , . . . , wN and let pi = P (cfw_wi ), i = 1, . . . , N. By axiom A1 we
1
Lecture Plan
Experiments, Outcomes and Events The Axioms of Probability Axiom consequences Finite Sample Space.
2
Experiments, Outcomes, Sample Space, and Events
Experiment: Toss a coin three times. Outcomes: The possible outcomes are hhh, hht, hth, h
(W4150 Intro to Probability and Statistics, S04) Sec. 5.3. (4) (a) P(X=2) = C52(3/4)2(1/4)3 = .0879 ; or you can use tables/Excel to find the same value by taking difference P(X=2) = P(X<2) P(X<1) = 20 b(x ;5,.75) - 10 b(x ;5,.75) (b) P(X<3) = 30 b(x ;5,.
(W4150 Intro to Probability and Statistics, S04) Sec. 6.4. (1) Use tables or built-in Excel function for normal distribution to answer each question here: (a) (b) (c) (d) (e) (f) Area = (1.43) = .9236 Area = 1 - (-.89) = .8133 Area = (-.65) - (-2.16) = .2
(W4150 Intro to Probability and Statistics, S04) Sec. 9.11. (1) (a) n = 200, p = 114 / 200 = .57, q = .43, z / 2 = z 0.02 = 2.05 Hence using the formula on p.258 we obtain the interval of the form: .57 2.05 (.57)(.43) / 200 < p < .57 + 2.05 (.57)(.43) / 2
(W4150 Intro to Probability and Statistics, S04) Sec. 10.4. (1) (a) (b) Sec.10.4. (2) (a) The training is effective (b) The training is effective Sec. 10.4. (3) (a) The firm is not guilty. (b) The firm is guilty. Sec.10.4 (6) (a) = Pcfw_to reject Ho when
(W4150 Intro to Probability and Statistics, S04) Sec. 10.12. (1) H 0 : p = .4 H1 : p > . 4 = .05 If we denote the number of those who choose lasagne by X then under H0 X ~ binomial(20;.4) and so P-value is given by P[X > 9 | p = .4] = 1 P[X < 8 | p = .4]
1
1.1
1.1.1
Introduction
Syllabus
Level of the course:
The course is given at an intermediate level. The course requires one year of calculus and a certain degree of mathematical maturity. This is an ambitious course in that we cover both probability and
Solutions to Homework #1
(W4150 Intro to Probability and Statistics, S04) Sec. 2.2. (1) (a) (b) (c) (d) (e) S = cfw_8,16,24,32,40,48 x2 + 4x 5 = (x + 5)(x 1) = 0 x = -5 and x =1, i.e. S = cfw_-5,1 S = cfw_T, HT, HHT, HHH S = cfw_N. America, S. America, Eu