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see those pdf and answer those questions in the pdf version. you need to use a computer to type the answer under those question.

2. Several experiments involving a fair deck of 52 cards are to be conducted.

(a) Cards will be repeatedly selected from the deck with replacement. Identify the

specific discrete probability model that best applies, and use it to set up BUT DO

NOT EVALUATE an expression for the probability that a random sample of

6 cards will contain exactly 4 Ã¢â‚¬Å“face cardsÃ¢â‚¬Â (i.e., Jack, Queen, or King). (4 pts)

(b) Cards will be repeatedly selected from the deck without replacement. Identify the specific

discrete probability model that best applies, and use it to set up BUT DO NOT EVALUATE

an expression for the probability that a random sample of 6 cards will contain exactly 4 Ã¢â‚¬Å“face

cardsÃ¢â‚¬Â (i.e., Jack, Queen, or King).

(4 pts)

(c) Cards will be repeatedly selected from the deck with replacement until exactly 4 Ã¢â‚¬Å“face cardsÃ¢â‚¬Â

(i.e., Jack, Queen, or King) are obtained. Identify the specific discrete probability model that

best applies, and use it to set up BUT DO NOT EVALUATE an expression for the probability

that a random sample of exactly 6 cards will be required.

(4 pts)

(d) Cards will be repeatedly selected from the deck with replacement until exactly 1 Ã¢â‚¬Å“face cardÃ¢â‚¬Â

(i.e., Jack, Queen, or King) is obtained. Identify the specific discrete probability model that

best applies, and use it to set up BUT DO NOT EVALUATE an expression for the probability

that a random sample of exactly 6 cards will be required.

(4 pts)

(e) Cards will be repeatedly selected from the deck without replacement until exactly 1 Ã¢â‚¬Å“face cardÃ¢â‚¬Â

(i.e., Jack, Queen, or King) is obtained. set up BUT DO NOT EVALUATE an expression for

the probability that a random sample of exactly 6 cards will be required.

(4 pts)

Ã¯Æ’â€¹ Numerically calculate your answer.

(2 pts)

ContinuedÃ¢â‚¬Â¦

(f) Suppose that in general, a population of finite size N units contains s Ã¢â‚¬Å“Successes.Ã¢â‚¬Â Units are to

be randomly drawn from the population without replacement. It can be formally proved that

the random variable X = Ã¢â‚¬Å“Number of trials until the first Success appearsÃ¢â‚¬Â has the pmf below.

Ã¯Æ’Â¦ N Ã¢Ë†â€™ xÃ¯Æ’Â¶

Ã¯Æ’Â§

Ã¯Æ’Â·

s Ã¢Ë†â€™1 Ã¯Æ’Â¸

Ã¯Æ’Â¨

p ( x) = P( X = x) =

Ã¯Æ’Â¦NÃ¯Æ’Â¶

Ã¯Æ’Â§ Ã¯Æ’Â·

Ã¯Æ’Â¨sÃ¯Æ’Â¸

Use this formula to recalculate the probability in (e), and show agreement of your answers. (3 pts)

(g) Extra Credit: Provide a rigorous mathematical proof of the formula in (f).

(5 pts)

3. Given any fixed value p, where 0 Ã¯â‚¬Â¼ p Ã¯â‚¬Â¼ 1 . Suppose that a continuous random variable X has the

piecewise linear pdf shown below.

Ã¯Æ’Â¬2

0Ã¯â€šÂ£ xÃ¯â‚¬Â¼ p

Ã¯Æ’Â¯ p x,

Ã¯Æ’Â¯

f ( x) = Ã¯Æ’Â

Ã¯Æ’Â¯ 2 ( x Ã¢Ë†â€™ 1), p Ã¯â€šÂ£ x Ã¯â‚¬Â¼ 1

Ã¯Æ’Â¯Ã¯Æ’Â® p Ã¢Ë†â€™ 1

2

and f ( x) = 0 otherwise.

p

(a) Prove that this is indeed a legitimate pdf, as claimed.

(3 pts)

(b) Determine the corresponding cdf F ( x) = P ( X Ã¯â€šÂ£ x) for all real x, and sketch its graph.

Clearly label all relevant features. Show all work.

(10 pts)

(c) Suppose a and b are two values, with 0 Ã¯â€šÂ£ a Ã¯â€šÂ£ p Ã¯â€šÂ£ b Ã¯â€šÂ£ 1 . Determine the interval probability

P (a Ã¯â€šÂ£ X Ã¯â€šÂ£ b) . Show all work.

(3 pts)

p +1

.

3

(8 pts)

(d) Show that the mean Ã¯ÂÂ =

(e) Determine the median Q2 . (Hint: There are two cases to consider.)

(6 pts)

4.

(a) Is it possible for a continuous function f ( x) to satisfy all of the following properties?

(i)

f ( x) Ã¯â€šÂ³ 0 for all x in some interval ( a, b)

(ii)

Ã¯Æ’Â²

Ã¯Æ’Â²

Ã¯Æ’Â²

(iii)

(iv)

b

a

b

a

b

a

(10 pts)

f ( x) dx = 1

x f ( x) dx = 2

x 2 f ( x) dx = 3

Hint: Why can f ( x) be viewed as the pdf of a continuous random variable X ?

(b) Answer part (a) if the last property were replaced by

be said about the random variable X?

Ã¯Æ’Â²

b

a

x 2 f ( x) dx = 4 . In this case, what can

(5 pts)

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