An electronics store has received a shipment of 20 table radios that have connections for an iPod or iPhone. Twelve of these have two slots (so they can accommodate both devices), and the other eight have a single slot. Suppose that six of the 20 radios are randomly selected to be stored under a shelf where the radios are displayed, and the remaining ones are placed in a storeroom. Let X= the number among the radios stored under the display shelf that have two slots
(a) What kind of a distribution does X have (name and values of all parameters)?
(b) Compute \( P(X = 2), P(X\leq2), and \hspace{2mm}P(X \geq 2) \).
(c) Calculate the mean value and standard deviation of X.
In: Statistics and Probability
In an assembly-line production of industrial robots, gearbox assemblies can be installed in one minute each if holes have been properly drilled in the boxes and in ten minutes if the holes must be redrilled. Twenty gearboxes are in stock, 2 with improperly drilled holes. Five gearboxes must be selected from the 20 that are available for installationcin the next five robots.
(a) Find the probability that all 5 gearboxes will fit properly.
(b) Find the mean, variance, and standard deviation of the time it takes to install these 5 gearboxes.
In: Statistics and Probability
A manufacturing company uses an acceptance scheme on items from a production line before they are shipped. The plan is a two-stage one. Boxes of 25 items are readied for shipment, and a sample of 3 items is tested for defectives. If any defectives are found, the entire box is sent back for 100% screening. If no defectives are found, the box is shipped.
(a) What is the probability that a box containing 3 defectives will be shipped?
(b) What is the probability that a box containing only 1 defective will be sent back for screening?
In: Statistics and Probability
Suppose that p = P(male birth) = 0.3, A couple wishes to have exactly two female children in their family. They will have children until this condition is fulfilled.
(a) What is the probability that the family has x male children?
(b) What is the probability that the family has four children?
(c) What is the probability that the family has at most four children?
(d) How many male children would you expect this family to have? How many children would you expect this family to have?
In: Statistics and Probability
Many utility companies promote energy conservation by offering discount rates to con-sumers who keep their energy usage below certain established subsidy standards. Arecent report notes that 70% of the people live in Phnom Penh have reduced their elec-tricity usage sufficiently to qualify for discounted rates. If five residential subscribers are randomly selected from Phnom Penh, find the probability of each of the following events:
(a) All five qualify for the favorable rates.
(b) At least four qualify for the favorable rates.
(c) At least two do not qualify the favorable rates.
In: Statistics and Probability
In a gambling game a person draws a single card from an ordinary 52-card playing deck. A person is paid $15 for drawing a jack or a queen and $5 for drawing a king or an ace. A person who draws any other card pays $4. If a person plays this game, what is the expected gain?
In: Statistics and Probability
[Markov inequality] If \( X \geq 0, i.e. X \) takes only nonnegative values, then for an \( a>0 \) we have \\( \hspace{3mm} P(X\geq a)\leq \frac{E(X)}{a} \)
In: Statistics and Probability
[Chebyshev inequality] For any random variable X and any \( a > 0 \), we have
\( P(|X-E(X)|\geq a)\leq\frac{V(X)}{a^2} \)
In: Statistics and Probability
Let X be a random variable with mean 11 and variance 9. Using Tchebysheff's theorem,find
(a) a lower bound for \( P(6 < X < 16) \)
(b) the value of C such that \( P(|X-11|\geq C)\leq 0.09 \)
In: Statistics and Probability