7.2 Problems on distribution and density functions  (Page 4/4)

Customers arrive at a service center with independent interarrival times in hours, which have exponential (3) distribution. The time X for the third arrival is thus gamma $(3,3)$ . Without using tables or m-programs, determine $P(X\le 2)$ .

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 $P(Z\le 20)$ .

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 variableuniformly 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 randomvariable 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?

Suppose $X\sim N(4,81)$ . That is, X has gaussian distribution with mean $\mu =4$ and variance ${\sigma }^2=81$ .

1. Use a table of standardized normal distribution to determine $P(2<X<8)$ and $P\left(\right|X-4|\le 5)$ .
2. Calculate the probabilities in part (a) with the m-function gaussian.

Suppose $X\sim N(5,81)$ . That is, X has gaussian distribution with $\mu =5$ and ${\sigma }^2=81$ . Use a table of standardized normal distribution to determine $P(3<X<9)$ and $P\left(\right|X-5|\le 5)$ . Check your results using the m-function gaussian.

Suppose $X\sim N(3,64)$ . That is, X has gaussian distribution with $\mu =3$ and ${\sigma }^2=64$ . Use a table of standardized normal distribution to determine $P(1<X<9)$ and $P\left(\right|X-3|\le 4)$ . Check your results with the m-function gaussian.

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 themean value μ .

The result of extensive quality control sampling shows that a certain model of digital watches coming off a production line have accuracy, in seconds per month,that is normally distributed with $\mu =5$ and ${\sigma }^2=300$ . To achieve a top grade, a watch must have an accuracy within the range of -5 to +10 secondsper month. What is the probability a watch taken from the production line to be tested will achieve top grade? Calculate, using a standardized normal table. Checkwith the m-function gaussian.

Use the m-procedure bincomp with various values of n from 10 to 500 and p from 0.01 to 0.7, to observe the approximation of the binomial distribution by the Poisson.

Use the m-procedure poissapp to compare the Poisson and gaussian distributions. Use various values of μ from 10 to 500.

Random variable X has density ${\displaystyle f_X\left(t\right)=\frac{3}{2}{t}^2,-1\le t\le 1}$ (and zero elsewhere).

1. Determine $P(-0.5\le X<0,8)$ , $P\left(\right|X|>0.5)$ , $P\left(\right|X-0.25|\le 0.5)$ .
2. Determine an expression for the distribution function.
3. Use the m-procedures tappr and cdbn to plot an approximation to the distribution function.

Random variable X has density function ${\displaystyle f_X\left(t\right)=t-\frac{3}{8}{t}^2,0\le t\le 2}$ (and zero elsewhere).

1. Determine $P(X\le 0.5)$ , $P(0.5\le X<1.5)$ , $P\left(\right|X-1|<1/4)$ .
2. Determine an expression for the distribution function.
3. Use the m-procedures tappr and cdbn to plot an approximation to the distribution function.

Random variable X has density function

1. Determine $P(X\le 0.5)$ , $P(0.5\le X<1.5)$ , $P\left(\right|X-1|<1/4)$ .
2. Determine an expression for the distribution function.
3. Use the m-procedures tappr and cdbn to plot an approximation to the distribution function.