7.3 Problems on Distribution and Density Functions

(See Exercises 3 and 4 from "Problems on Random Variables and Probabilities"). The class $\left\{C_{j} : 1 \leq j \leq 10\right\}$ is a partition. Random variable X has values $\left\{1 , 3 , 2 , 3 , 4 , 2 , 1 , 3 , 5 , 2\right\}$ on C₁ through C₁₀, respectively, with probabilities 0.08, 0.13, 0.06, 0.09, 0.14, 0.11, 0.12, 0.07, 0.11, 0.09. Determine and plot the distribution function F_X.

(See Exercise 6 from "Problems on Random Variables and Probabilities"). A store has eight items for sale. The prices are $3.50, $5.00, $3.50, $7.50, $5.00, $5.00, $3.50, and $7.50, respectively. A customer comes in. She purchases one of the items with probabilities 0.10, 0.15, 0.15, 0.20, 0.10 0.05, 0.10 0.15. The random variable expressing the amount of her purchase may be written

Determine a…

(See Exercise 12 from "Problems on Random Variables and Probabilities"). The class $\left\{A , B , C , D\right\}$ has minterm probabilities

Determine and plot the distribution function for the random variable $X = I_{A} + I_{B} + I_{C} + I_{D}$, which counts the number of the events which occur on a trial.

Suppose a is a ten digit number. A wheel turns up the digits 0 through 9 with equal probability on each spin. On ten spins what is the probability of matching, in order, k or more of the ten digits in a, $0 \leq k \leq 10$? Assume the initial digit may be zero.

In a thunderstorm in a national park there are 127 lightning strikes. Experience shows that the probability of of a lightning strike starting a fire is about 0.0083. What is the probability that k fires are started, $k = 0 , 1 , 2 , 3$?

A manufacturing plant has 350 special lamps on its production lines. On any day, each lamp could fail with probability $p = 0 . 0017$. These lamps are critical, and must be replaced as quickly as possible. It takes about one hour to replace a lamp, once it has failed. What is the probability that on any day the loss of production time due to lamp failaures is k or fewer hours, $k = 0 , 1 , 2 , 3 , 4 , 5 ?$

Two hundred persons buy tickets for a drawing. Each ticket has probability 0.008 of winning. What is the probability of k or fewer winners, $k = 2 , 3 , 4 ?$

Two coins are flipped twenty times. What is the probability the results match (both heads or both tails) k times, $0 \leq k \leq 20$?

Thirty members of a class each flip a coin ten times. What is the probability that at least five of them get seven or more heads?

For the system in the exercise, call a day in which one or more failures occur among the 350 lamps a “service day.” Since a Bernoulli sequence “starts over” at any time, the sequence of service/nonservice days may be considered a Bernoulli sequence with probability p₁, the probability of one or more lamp failures in a day.

1. Beginning on a Monday morning, what is the probability the first service da…

For the system in the exercise and the exercise assume the plant works seven days a week. What is the probability the third service day occurs by the end of 10 days? Solve using the negative binomial distribution; repeat using the binomial distribution.

A residential College plans to raise money by selling “chances” on a board. Fifty chances are sold. A player pays $10 to play; he or she wins $30 with probability $p = 0 . 2$. The profit to the College is

Determine the distribution for X and calculate $P \left(X > 0\right)$, $P \left(X \geq 200\right)$, and

$P \left(X \geq 300\right)$.

A single six-sided die is rolled repeatedly until either a one or a six turns up. What is the probability that the first appearance of either of these numbers is achieved by the fifth trial or sooner?

Consider a Bernoulli sequence with probability $p = 0 . 53$ of success on any component trial.

1. The probability the fourth success will occur no later than the tenth trial is determined by the negative binomial distribution. Use the procedure nbinom to calculate this probability .
2. Calculate this probability using the binomial distribution.

Fifty percent of the components coming off an assembly line fail to meet specifications for a special job. It is desired to select three units which meet the stringent specifications. Items are selected and tested in succession. Under the usual assumptions for Bernoulli trials, what is the probability the third satisfactory unit will be found on six or fewer trials?

The number of cars passing a certain traffic count position in an hour has Poisson (53) distribution. What is the probability the number of cars passing in an hour lies between 45 and 55 (inclusive)? What is the probability of more than 55?

Suppose $X \sim$ binomial (12, 0.375), $Y \sim$ Poisson (4.5), and $Z \sim$ exponential (1/4.5). For each random variable, calculate and tabulate the probability of a value at least k, for integer values $3 \leq k \leq 8$.

The number of noise pulses arriving on a power circuit in an hour is a random quantity having Poisson (7) distribution. What is the probability of having at least 10 pulses in an hour? What is the probability of having at most 15 pulses in an hour?

The number of customers arriving in a small specialty store in an hour is a random quantity having Poisson (5) distribution. What is the probability the number arriving in an hour will be between three and seven, inclusive? What is the probability of no more than ten?

Random variable $X \sim$ binomial (1000, 0.1).

1. Determine $P \left(X \geq 80\right) , P \left(X \geq 100\right) , P \left(X \geq 120\right)$
2. Use the appropriate Poisson distribution to approximate these values.

The time to failure, in hours of operating time, of a televesion set subject to random voltage surges has the exponential (0.002) distribution. Suppose the unit has operated successfully for 500 hours. What is the (conditional) probability it will operate for another 500 hours?

For $X \sim$ exponential $\left(\lambda\right)$, determine $P \left(X \geq 1 / \lambda\right)$, $P \left(X \geq 2 / \lambda\right)$.

Twenty “identical” units are put into operation. They fail independently. The times to failure (in hours) form an iid class, exponential (0.0002). This means the “expected” life is 5000 hours. Determine the probabilities that at least k, for $k = 5 , 8 , 10 , 12 , 15$, will survive for 5000 hours.

Five people wait to use a telephone, currently in use by a sixth person. Suppose time for the six calls (in minutes) are iid, exponential (1/3). What is the distribution for the total time Z from the present for the six calls? Use an appropriate Poisson distribution to determine $P \left(Z \leq 20\right)$.

A random number generator produces a sequence of numbers between 0 and 1. Each of these can be considered an observed value of a random variable uniformly distributed on the interval [0, 1]. They assume their values independently. A sequence of 35 numbers is generated. What is the probability 25 or more are less than or equal to 0.71? (Assume continuity. Do not make a discrete adjustment.)

Five “identical” electronic devices are installed at one time. The units fail independently, and the time to failure, in days, of each is a random variable exponential (1/30). A maintenance check is made each fifteen days. What is the probability that at least four are still operating at the maintenance check?

Items coming off an assembly line have a critical dimension which is represented by a random variable $\sim$ N(10, 0.01). Ten items are selected at random. What is the probability that three or more are within 0.05 of the mean value μ.