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Exam 4: Discrete Random Variables

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Question 1

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A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the probability for the value of x = 5. $\begin{array} { c | c c c c c } \mathrm { x } & 2 & 3 & 5 & 8 & 10 \\\hline \mathrm { p } ( \mathrm { x } ) & 0.10 & 0.20 & ? ? ? & 0.30 & 0.10\end{array}$

A) 0.7
B) 0.1
C) 0.3
D) 0.2

E) C) and D)
F) B) and D)

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Question 2

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Compute $\left( \begin{array} { l } 4 \\4\end{array} \right)$

A) 1
B) 4
C) 6
D) 16

E) B) and D)
F) A) and D)

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Question 3

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The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 4 goals per game. Find the probability that the team will score more than three goals during a game.

We believe that 81% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 39 students from the population. How many of the sampled students do we expect to consider statistics to be an exciting subject?

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 42% of all employees believe their company president possesses low ethical standards. Assume that responses were randomly and independently collected. A president of a local company that employs 1,000 people does not believe the paper's claim applies to her company. If the claim is true, how many of her company's employees believe that she possesses low ethical standards?

Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint. For this weekend's big project, the man randomly selects four of these 1-gallon cans to paint his mother's living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the standard deviation of this distribution.

As part of a promotion, both you and your roommate are given free cellular phones from a batch of 13 phones. Unknown to you, four of the phones are faulty and do not work. Find the probability that one of the two phones is faulty.

Given that x is a hypergeometric random variable, compute p(x) for N = 6, n = 3, r = 3, and x = 1.

Mamma Temte bakes six pies each day at a cost of $2 each. On 39% of the days she sells only two pies. On 38% of the days, she sells 4 pies, and on the remaining 23% of the days, she sells all six pies. If Mama Temte sells her pies for $4 each, what is her expected profit for a day's worth of pies? [Assume that any leftover pies are given away.]

A small life insurance company has determined that on the average it receives 3 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.

Classify the following random variable according to whether it is discrete or continuous. The number of goals scored in a soccer game

A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution below: $\begin{array} { l | c c c } \text { Price } & \$ 10.00 & \$ 12.50 & \$ 15.00 \\\hline \text { Probability } & 0.4 & 0.45 & 0.15\end{array}$ How much should the lab budget for next year's frog orders assuming this distribution does not change? (Hint: Find the expected price and assume 52 weeks per year.)

A local newspaper claims that 70% of the items advertised in its classifieds section are sold within 1 week of the first appearance of the ad. To check the validity of the claim, the newspaper randomly selected n = 25 advertisements from last year's classifieds and contacted the people who placed the ads. They found that 16 of the 25 items sold within a week. Based on the newspaper's claim, is it likely to observe $x \leq 16$ who sold their item within a week? Use a binomial probability table.

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 5. Find the probability that exactly six babies will be born during a particular 1-hour period at this hospital.

Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint. For this weekend's big project, the man randomly selects four of these 1-gallon cans to paint his mother's living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the probability that at least one of the four cans selected contains an incorrect mix of paint.

Calculate the mean for the discrete probability distribution shown here. $\begin{array} { l | c c c c } X & 2 & 3 & 8 & 10 \\\hline P ( X ) & .2 & .3 & .3 & .2\end{array}$

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the standard deviation of the distribution. $\begin{array} { c | c c c c c } \mathrm { x } & 2 & 3 & 5 & 8 & 10 \\\hline \mathrm { p } ( \mathrm { x } ) & 0.10 & 0.20 & 0.30 & 0.30 & 0.10\end{array}$

Management at a home improvement store randomly selected 95 customers and observed their shopping habits. They recorded the number of items each of the customers purchased as well as the total time the customers spent in the store. Identify the types of variables recorded by the managers of the home improvement store.

A bank offers online banking to its customers free of charge. While online, customers can also sign up for additional services that the bank offers. Let x be the number of customers who sign up for additional services online each day. Suppose the distribution of x is approximated well by a Poisson distribution with mean λ = 42.3. Find E(x) and interpret its value.

Classify the following random variable according to whether it is discrete or continuous. The blood pressures of a group of students the day before the final exam