A student is going to take an survey of ten problems. Suppose among the problems that may appear in the exam, the student can solve 20% of them correctly, and 60% partially correctly. If you know that the student can solve exactly three problems in the survey correctly, what is the expected number of problems that he or she can solve partially correctly ?

Find the conditional probability that the student can solve exactly three problems partially correctly given that he or she can solve three problems correctly.

Problem 13-14 Value-at-Risk (VaR) Statistic (LO4, CFA6)

a. A stock has an annual return of 11.4 percent and a standard deviation of 58 percent. What is the smallest expected gain over the next year with a probability of 5 percent? (Do not round intermediate calculations. Round the z-score value to 3 decimal places when calculating your answer. Enter your answer as a percent rounded to 2 decimal places.)

Smallest Expected Gain __________%

b. Does this number make sense?

• Yes

• No

1. Consider the time dependent Hamiltonian H=H0+W(t). |φn is a basis of H₀. When t<0, the system is in one of the eigen states, |φi, of the unperturbed Hamiltonian H₀. At ?=0 an, a time dependent perturbation Wt=Ccos(ωt) is applied to the system for a sufficiently long time, here C is a complex number and w is the angular frequency of the applied perturbation. Calculate the probability of finding the system in |φf eigen states of H₀ when the system in resonance ω≈ωfi. Assume also that Ef-Eiℏ=ωfi >0.

Grade:ABCDF

Probability:0.10.30.40.10.1

To calculate student grade point averages, grades are expressed in a numerical scale with A = 4, B = 3, and so on down to F = 0.

Find the expected value. This is the average grade in this course.

Explain how to simulate choosing students at random and recording their grades. Simulate 50 students and find the mean of their 50 grades. Compare this estimate of the expected value with the exact expected value from part (a). (The law of large numbers says that the estimate will be very accurate if we simulate a very large number of students.)

A worn machine is known to produce 10% defective components. If the random variable X is the number of defective components produced in a run pf 3 components, find the probabilities that X takes the values 0 to 3.

Suppose now that a similar machine which is known to produce 1% defective components is used for a production run of 40 components.We wish to calculate the probability that two defective items are produced. Essentially we are assuming thatX~B(40,0.01) and we use both the binomial distribution and its Poisson approxiamation for comparison.

1. A coin is tossed 3 times. Let x be the random discrete variable representing the number of times tails comes up.

a) Create a sample space for the event;   

b) Create a probability distribution table for the discrete variable x;                

c) Calculate the expected value for x.

2. For the data below, representing a sample of times (in minutes) students spend solving a certain Statistics problem, find P₃₅, range, Q₂ and IQR.

3.0, 3.2, 4.6, 5.2 3.2, 3.5

=> no handwriting. I can't read it correctly.

Some job applicants are required to have several interviews before a decision is made. The number of required interviews and the corresponding probabilities are: 1 (0.09); 2 (0.31); 3 (0.37); 4 (0.12); 5 (0.05); 6 (0.05).

a) Does this information describe a probability distribution? What is the sum of probabilities?

b) Assuming it does, find its mean and standard deviation.

c) Use the range rule of thumb to identify the range of values for usual numbers of interviews.

d) Is it unusual to have a decision after just one interview? Explain.

Suppose that in a certain region of California, earthquakes occur at the average rate of 7 per year.

(a) What is the probability that in exactly three of the next eight years, no earthquakes occur?

(b) What is the **expect**ed number of years to wait until we have a year with exactly 7 earthquakes?

(c) In the next century, how many years would you **expect** to see with more than 10 earthquakes?

Hint: When you see the word "expect" you should expect to use the expected value!

An online account password for a certain website consists of eight characters, where at least one must be a digit (i.e. a number from 0-9).

a. How many different passwords are possible if only lowercase letters and digits can be used?

b. How many different passwords are possible if a user wants to include single capital letter somewhere in their password?

c. If a computer program randomly generates eight characters (such that each could be either a digit or any lowercase letter), what is the probability that a valid password is generated?