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Suppose that X is a random variable with mean and variance both equal to 20. What can be said about P{0 < X < 40}?

The number of automobiles sold weekly at a certain dealership is a random variable with an expected value of$16$. Give an upper bound to the probability that

$\left(a\right)$next week鈥檚 sales exceed$18$;

$\left(b\right)$next week鈥檚 sales exceed$25$.

It $X$has a variance${蟽}^2$, then 蟽, the positive square root of the variance, is called the standard deviation. It $X$has to mean $渭$and standard deviation$蟽$, to show that$P\left\{\right|X-渭|鈮�k蟽\}鈮�\frac{1}{k^2}.$

A tobacco company claims that the amount of nicotine in one of its cigarettes is a random variable with a mean of $2.2$mg and a standard deviation of $0.3$mg. However, the average nicotine content of $100$randomly chosen cigarettes was $3.1$mg. Approximate the probability that the average would have been as high as or higher than $3.1$ if the company鈥檚 claims were true

Civil engineers believe that W, the amount of weight (in units of $1000$pounds) that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with a mean of $400$and standard deviation of$40$. Suppose that the weight (again, in units of $1000$pounds) of a car is a random variable with a mean of $3$and standard deviation$.3$. Approximately how many cars would have to be on the bridge span for the probability of structural damage to exceed$.1$?

Each of the batteries in a collection of $40$batteries is equally likely to be either a type A or a type B battery. Type A batteries last for an amount of time that has a mean of $50$and a standard deviation of $15$; type B batteries last for a mean of $30$and a standard deviation of 6.

(a) Approximate the probability that the total life of all $40$batteries exceeds $1700$

(b) Suppose it is known that $20$of the batteries are type A and $20$are type B. Now approximate the probability that the total life of all $40$batteries exceeds $1700.$

A clinic is equally likely to have 2, 3, or 4 doctors volunteer for service on a given day. No matter how many volunteer doctors there are on a given day, the numbers of patients seen by these doctors are independent Poisson random variables with a mean of $30$. Let X denote the number of patients seen in the clinic on a given day.

(a) Find $E\left[X\right]$

(b) Find Var$\left(X\right)$

(c) Use a table of the standard normal probability distribution to approximate P$P\{X>65\}$.

We have $100$components that we will put to use in a sequential fashion. That is, the component $1$is initially put in use, and upon failure, it is replaced by a component$2$, which is itself replaced upon failure by a componentlocalid="1649784865723" $3$, and so on. If the lifetime of component i is exponentially distributed with a mean $10+i/10,i=1,...,100$estimate the probability that the total life of all components will exceed$1200$. Now repeat when the life distribution of component i is uniformly distributed over$(0,20+i/5),i=1,...,100$.

The Chernoff bound on a standard normal random variable$Z$gives$P\{Z>a\}鈮�e^{-a^2/2},a>0$. Show, by considering the density$Z$, that the right side of the inequality can be reduced by the factor$2$. That is, show that

$P\{Z>a\}鈮�\frac{1}{2}{e}^{-a^2/2}a>0$

The strong law of large numbers states that with probability 1, the successive arithmetic averages of a sequence of independent and identically distributed random variables converge to their common mean . What do the successive geometric averages converge to? That is, what is

$\underset{n鈫�鈭�}{\lim }{\left(\underset{i=1}{\overset{n}{鈭�}}{X}_i\right)}^{1/n}$