Miles  Freq
 0-4     3
 5-9    14
10-14   13
15-19    4

Select the most appropriate sentence corresponding to two standard deviations.

*About 68% of students drive between 5.5212 miles and 13.7730 miles to somewhere
*At least 88.9% of students drive between -2.7306 miles and 22.0248 miles to 
*About 99.7% of students drive between 1.3953 miles and 17.8989 miles to 
*About 68% of students drive less than 22.0248 miles to
*About 95% of students drive between 5.5212 miles and 13.7730 miles to
*About 99.7% of students drive between -2.7306 miles and 22.0248 miles to
*About 68% of students drive between 1.3953 miles and 17.8989 miles to
*About 99.7% of students drive between 5.5212 miles and 13.7730 miles to
*At least 75% of students drive between -2.7306 miles and 22.0248 miles to
*At least 75% of students drive less than 22.0248 miles to
*About 95% of students drive between 1.3953 miles and 17.8989 miles to
*At least 75% of students drive between 1.3953 miles and 17.8989 miles to
*About 95% of students drive less than 22.0248 miles to 
*At least 88.9% of students drive between 1.3953 miles and 17.8989 miles to
*About 99.7% of students drive less than 22.0248 miles to
*About 95% of students drive between -2.7306 miles and 22.0248 miles to
*At least 88.9% of students drive less than 22.0248 miles to
*About 68% of students drive between -2.7306 miles and 22.0248 miles to

A class has 40 students.

• Thirty students are prepared for the exam,

• Ten students are unprepared. The professor writes an exam with 10 questions, some are hard and some are easy.

• 7 questions are easy. Based on past experience, the professor knows that: – Prepared students have a 90% chance of answering easy questions correctly – Unprepared students have a 50% chance of answering easy questions correctly.

• 3 questions are hard. Based on past experience, the professor knows that: – Prepared students have a 50% chance of answering hard questions correctly – Unprepared students have a 10% chance of answering hard questions correctly

• Each student’s performance on each question is independent of their performance on other questions.

(a) Find the probability that a prepared student answers all 10 questions correctly.

(b) What is the probability that at least one of the 30 prepared students answers all 10 questions correctly. Assume that each student’s score is independent of every other student.

(c) Let P be the number of questions answered correctly by a randomly chosen prepared student, and let U be the number answered correctly by a randomly chosen unprepared student. Find E[P] and E[U]

(d) Find Var(P) and Var(U)

A class has 40 students.

Thirty students are prepared for the exam, • Ten students are unprepared.

The professor writes an exam with 10 questions, some are hard and some are easy.

• 7 questions are easy. Based on past experience, the professor knows that:

– Prepared students have a 90% chance of answering easy questions correctly

– Unprepared students have a 50% chance of answering easy questions correctly.

• 3 questions are hard. Based on past experience, the professor knows that:

– Prepared students have a 50% chance of answering hard questions correctly

– Unprepared students have a 10% chance of answering hard questions correctly

• Each student’s performance on each question is independent of their performance on other questions.

(a) Find the probability that a prepared student answers all 10 questions correctly.

(b) What is the probability that at least one of the 30 prepared students answers all 10 questions correctly. Assume that each student’s score is independent of every other student.

(c) Let P be the number of questions answered correctly by a randomly chosen prepared student, and let U be the number answered correctly by a randomly chosen unprepared student. Find E[P] and E[U]

Let L be a linear map between linear spaces U and V, such that L: U -> V and let l_{ij} be the matrix associated with L w.r.t bases {u_i} and {v_i}. Show l_{ij} changes w.r.t a change of bases (i.e u_i -> u'_i and v_j -> v'_j)

A long conducting pipe has a rectangular cross section with sides of lengths a and b. One face of the pipe is maintained at a constant potential V = V0 while the other 3 faces are grounded (V = 0). Using separation of variables, find the potential for points inside the pipe V (x,y).

A laptop computer that costs $1200 new has a book value of $425 after 2 years. (a) Find a linear model V = mt + b. V(t) = Incorrect: Your answer is incorrect. (b) Find an exponential model V = aekt. (Round your values to four decimal places.)

Let Σ be a finite alphabet with n letters and let R be the relation on Σ* defined as follows: R = {(u, v): every letter in u occurs somewhere in v, and every letter in v occurs somewhere in u} Then R is an equivalence relation with exactly 2n equivalence classes.

T or F?

Problem 3. An isometry between inner-product spaces V and W is a linear
operator L in B (V ,W) that preserves norms and inner-products. If x, y in V
and if L is an isometry, then we have <L(x),L(y)>_W = <x, y>_V .
Suppose that V and W are both real, n-dimensional inner-product spaces.
Thus the scalar field for both is R and both of them have a basis consisting of
n elements. Show that V and W are isometric by demonstrating an isometry
between them.
Hint: take both bases, and cite some linear algebra result that says that
you can orthonormalize them. Prove (or cite someone to convince me) that you
can define a linear function by specifying its action on a basis. Finally, define
your isometry by deciding what it should do on an orthonormal basis for V , and
prove that it preserves inner-products (and thus norms).

The joint probability distribution of variables X and Y is shown in the table below.

...............................................................................X.......................................................................

1. Calculate E(XY)

2. Determine the marginal probability distributions of X and Y.

3. Calculate E(X) and E(Y)

4. Calculate V(X) and V(Y)

5. Are X and Y independent? Explain.

6. Find P(Y = 2| X = 1)

7. Calculate COV(X,Y). Did you expect this answer? Why?

8. Find the probability distribution of the random variable X + Y.     

i.   Calculate E(X + Y) directly by using the probability distribution of X + Y.

1. Calculate V(X + Y) directly by using the probability distribution of X + Y, and verify that V(X + Y) = V(X) + V(Y). Did you expect this result? Why?

A particle with positive charge q = 9.61 10-19 C moves with a velocity v = (3î + 4ĵ − k) m/s through a region where both a uniform magnetic field and a uniform electric field exist. (a) Calculate the total force on the moving particle, taking B = (4î + 3ĵ + k) T and E = (3î − ĵ − 4k) V/m. (Give your answers in N for each component.) Fx = N Fy = N Fz = N (b) What angle does the force vector make with the positive x-axis? (Give your answer in degrees counterclockwise from the +x-axis.) ° counterclockwise from the +x-axis (c) What If? For what vector electric field would the total force on the particle be zero? (Give your answers in V/m for each component.) Ex = V/m Ey = V/m Ez = V/m