Math Answers

A travel agency receives an average of 150 calls per hour (time between calls are exponentially distributed). It takes an operator an average of 5 minutes to handle a call (exponentially distributed). If a caller gets a busy signal, the travel agency assumes that he or she will call a competitor, and the travel agency will lose an average of $50 in profit. The cost of keeping a phone line open is $12 per hour. How many operators should the travel agency have on duty?

A store donated a lot of 8 computer sets that includes 3 which are malfunctioning or defective. If 4 of this computer sets are chosen at random for delivery to a school.

i) What will be the probability mass function of a random variable 𝑌?

ii) What will be the expected value of 𝑌?

For each fixed λ > 0, let X have a Poisson distribution with parameter λ. Suppose λ

itself is a random variable with the gamma distribution

f(λ) =







1

Γ(n)

λ

n−1

e

−λ

, λ ≥ 0

0, λ < 0

where n is a fixed positive constant. Show that

P(X = k) = Γ(k + n)

Γ(n)Γ(k + 1) 

1

2

k+n

, k = 0, 1, 2

Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.

a) ∀x(C(x) → F(x))

b) ∀x(C(x) ∧ F(x))

c) ∃x(C(x) → F(x))

d) ∃x(C(x) ∧ F(x))

Determine the lowest common multiple LCM of the denominator of the fractions 5/9 and 3/ 15

Present an argument in one or two sentences whether you think there is likely to

be, or not likely to be, a relationship between the following pairs of variables. In each case, if

you think there is no relationship, explain why not; but if you think there is a relationship,

explain whether and why you think it is a positive or negative relationship. (Note: the

explanation is more important than the yes/no, +/- part of your answer.)

Q1.1 Childhood mortality and GPC per capita

Q1.2 Childhood mortality and Literacy

Q1.3 GPC per capita and Literacy

Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x) = sin(x),    0 ≤ x < 𝜋

2

Five thousand lottery tickets are sold for $1 each. One ticket will win $1000, two tickets will win $500 each, and ten tickets will win $100 each. Let X denote the net gain from the purchase of a randomly selected ticket.

a. Construct the probability distribution of C

b. Compute the standard deviation of X

2ydx+x(x^2lny-1)dy=0

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) = At^pe^−kt

 (where t > 0) is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A = 0.03, p = 4, k = 0.05, and t is measured in minutes, estimate the times t corresponding to the inflection points. (Round your answers to two decimal places.)t = min (smaller value) t = min (larger value)