2. A hospital receives 90% of its Covid-19 vaccine from Company Red and
the remainder from Company Blue.
A hospital tests n = 15 randomly selected vials from one shipment for
their effectiveness.
(a) (5 points) Compute the probability that exactly 3 of these 15 vials
are ineffective if this shipment comes from Company Red. (No need
to simplify.)
(b) (5 points) Compute the probability that exactly 3 of these 15 vials
are ineffective if this shipment comes from Company Blue. (No need
to simplify.)
(c) (5 points) Compute the probability that exactly 3 of these 15 vials
are ineffective. (No need to simplify.)
(d) (5 points) Compute the conditional probability that this shipment
come from Company Red, given that exactly three of these 15 vials
are ineffective. (No need to simplify.)
3. Friendly Instructor Cereal began placing one of the 250 Pokemon figurines
randomly into each cereal box they sold.
(a) (5 points) On the average, how many boxes of cereal should you buy
until you have three copies of the Pikachu figurine?
(b) (10 points) Approximate the probability that there are no Pikachu
figurines at all in 500 boxes.
.(c) (10 points) On the average, how many boxes of cereal should you buy
until you obtain a complete collection consisting of all the pokemons?
Give an approximate answer. (Hint: use Taylor’s expansion)
4. Let X be the random variable with moment generating function
M(t) = e2t+50t2
(a) (5 points) Calculate P(−8 X 22).
(b) (5 points) Calculate P(|X − 2| >
p384.1).
5. Let X be a continuous random variable with probability density function
f(x) = 2x for 0 x 1.
Let N be a fixed nonnegative real number, and let Y be a random variable
given by
Y =
(
3N if X N;
4X − N if X < N.
(a) (10 points) Prove that E[Y ] = −4
3 N3 + 3N for N 1. (Explain
your reasoning clearly)
(b) (10 points) Prove that E[Y ] = 83
− N for N 1. (Explain your
reasoning clearly)
(c) (10 points) Find the value of N that maximizes E[Y ]. (Explain your
reasoning clearly.)
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