Exercises

1.
Consider a normal random variable with mean 50 and standard deviation 10, and a random sample of size 80 from which we are to compute the values of $\bar{{X}}$, the sample mean. What is the probability of getting a value of $\bar{{X}}$ as low as 46?BITMAPSETAnswer0.2214in0.205in0ina1

2.
Suppose a working widget deteriorates very little with age. That is, a widget that has been running for some time will have nearly the same failure probability during the following hour as it had during its first hour of operation. Then, the failure times have an exponential distribution P(Tt) of the form 1 - e-$\scriptstyle {\frac{{x}}{{\mu }}}$. Given that the widget has a mean life of 5 years, what is the probability that the widget will have a lifetime exceeding 7.5 years? If the widget is guaranteed for 2 years, what percentage of such widgets can be expected to need replacement while under warranty?BITMAPSETAnswer0.2214in0.205in0ina2

3.
A widget has a mean life of 5 years with a standard deviation of 2 years. Assuming a normal distribution, what is the probability that the widget will have a lifetime exceeding 7.5 years? If the widget is guaranteed for 2 years, what percentage of such widgets can be expected to need replacement while under warranty?BITMAPSETAnswer0.2214in0.205in0ina3

4.
The mean of a continuous distribution with probability density function f (u) is the integral $\int_{{-\infty }}^{{\infty
}}$uf (u)du = μ of the product of the variable and the probability density function. The variance is the integral $\int_{{-\infty }}^{{\infty
}}$$\left(\vphantom{
u-\mu }\right.$u - μ$\left.\vphantom{
u-\mu }\right)^{{2}}_{}$f (u)du. Find the mean and variance for each of the continuous distributions discussed in this chapter.BITMAPSETAnswer0.2214in0.205in0ina4

5.
The mean of a discrete distribution with probability density function f (u) is the sum $\sum_{{-\infty }}^{{\infty }}$uf (u) = μ, and the variance is $\sum_{{-\infty }}^{{\infty }}$$\left(\vphantom{
u-\mu }\right.$u - μ$\left.\vphantom{
u-\mu }\right)^{{2}}_{}$f (u) = σ2.

6.
A die is cast until a 4 appears. What is the probability that it must be cast more than five times?

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7.
A telephone switchboard handles 600 calls on average during a single rush hour. The board can make a maximum of 20 connections per minute. Use the Poisson distribution to evaluate the probability that the board will be overtaxed during any given minute of a rush hour. BITMAPSETAnswer0.2214in0.205in0ina7

8.
Find the probability that x2≤4 for a normal distribution with mean 1 and standard deviation 1.

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