Chapter 5

The width of a casing for a door is normally distributed with a mean of 24 inches and a standard deviation of \(1 / 8\) inch. The width of a door is normally distributed with a mean of \(237 / 8\) inches and a standard deviation of \(1 / 16\) inch. Assume independence. (a) Determine the mean and standard deviation of the difference between the width of the casing and the width of the door. (b) What is the probability that the width of the casing minus the width of the door exceeds \(1 / 4\) inch? (c) What is the probability that the door does not fit in the casing?

An article in Knee Surgery Sports Traumatology, Arthroscopy ["Effect of Provider Volume on Resource Utilization for Surgical Procedures" (2005, Vol. 13, pp. \(273-279\) ) showed a mean time of 129 minutes and a standard deviation of 14 minutes for ACL reconstruction surgery for high-volume hospitals (with more than 300 such surgeries per year). If a high-volume hospital needs to schedule 10 surgeries, what are the mean and variance of the total time to complete these surgeries? Assume that the times of the surgeries are independent and normally distributed.

An automated filling machine fills soft-drink cans, and the standard deviation is 0.5 fluid ounce. Assume that the fill volumes of the cans are independent, normal random variables. (a) What is the standard deviation of the average fill volume of 100 cans? (b) If the mean fill volume is 12.1 oz, what is the probability that the average fill volume of the 100 cans is less than 12 oz? (c) What should the mean fill volume equal so that the probability that the average of 100 cans is less than 12 oz is \(0.005 ?\) (d) If the mean fill volume is 12.1 oz, what should the standard deviation of fill volume equal so that the probability that the average of 100 cans is less than 12 oz is \(0.005 ?\)

The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Assume that the thickness is normally distributed and that the thicknesses of different wafers are independent. (a) Determine the probability that the average thickness of 10 wafers is either greater than 11 or less than 9 micrometers. (b) Determine the number of wafers that need to be measured such that the probability that the average thickness exceeds 11 micrometers is 0.01 (c) If the mean thickness is 10 micrometers, what should the standard deviation of thickness equal so that the probability that the average of 10 wafers is either more than 11 or less than 9 micrometers is \(0.001 ?\)

Assume that the weights of individuals are independent and normally distributed with a mean of 160 pounds and a standard deviation of 30 pounds. Suppose that 25 people squeeze into an elevator that is designed to hold 4300 pounds. (a) What is the probability that the load (total weight) exceeds the design limit? (b) What design limit is exceeded by 25 occupants with probability \(0.0001 ?\)

Weights of parts are normally distributed with variance \(\sigma^{2}\). Measurement error is normally distributed with mean 0 and variance \(0.5 \sigma^{2}\), independent of the part weights, and adds to the part weight. Upper and lower specifications are centered at \(3 \sigma\) about the process mean. (a) Without measurement error, what is the probability that a part exceeds the specifications? (b) With measurement error, what is the probability that a part is measured as being beyond specifications? Does this imply it is truly beyond specifications? (c) What is the probability that a part is measured as being beyond specifications if the true weight of the part is \(1 \sigma\) below the upper specification limit?

Three electron emitters produce electron beams with changing kinetic energies that are uniformly distributed in the ranges \([3,7],[2,5],\) and \([4,10] .\) Let \(Y\) denote the total kinetic energy produced by these electron emitters. (a) Suppose that the three beam energies are independent. Determine the mean and variance of \(Y\). (b) Suppose that the covariance between any two beam energies is \(-0.5 .\) Determine the mean and variance of \(Y\). (c) Compare and comment on the results in parts (a) and (b).

In Exercise \(5-31,\) the monthly demand for MMR vaccine was assumed to be approximately normally distributed with a mean and standard deviation of 1.1 and 0.3 million doses, respectively. Suppose that the demands for different months are independent, and let \(Z\) denote the demand for a year (in millions of does). Determine the following: (a) Mean, variance, and distribution of \(Z\) (b) \(P(Z<13.2)\) (c) \(P(11

The rate of return of an asset is the change in price divided by the initial price (denoted as \(r\) ). Suppose that \(\$ 10,000\) is used to purchase shares in three stocks with rates of returns \(X_{1}, X_{2}, X_{3}\). Initially, \(\$ 2500, \$ 3000,\) and \(\$ 4500\) are allocated to each one, respectively. After one year, the distribution of the rate of return for each is normally distributed with the following parameters: \(\mu_{1}=0.12, \sigma_{1}=0.14, \mu_{2}=0.04, \sigma_{2}=0.02, \mu_{3}=0.07, \sigma_{3}=0.08\) (a) Assume that these rates of return are independent. Determine the mean and variance of the rate of return after one year for the entire investment of \(\$ 10,000\). (b) Assume that \(X_{1}\) is independent of \(X_{2}\) and \(X_{3}\) but that the covariance between \(X_{2}\) and \(X_{3}\) is \(-0.005 .\) Repeat part (a). (c) Compare the means and variances obtained in parts (a) and (b) and comment on any benefits from negative covariances between the assets.

Let \(X\) be a binomial random variable with \(p=0.25\) and \(n=3\). Determine the probability distribution of the random variable \(Y=X^{2}\)

Suppose that \(X\) is a continuous random variable with probability distribution $$ f_{X}(x)=\frac{x}{18}, \quad 0 \leq x \leq 6 $$ (a) Determine the probability distribution of the random variable \(Y=2 X+10\). (b) Determine the expected value of \(Y\).

A random variable \(X\) has the probability distribution $$ f_{X}(x)=e^{-x}, \quad x \geq 0 $$ Determine the probability distribution for the following: (a) \(Y=X^{2}\) (b) \(Y=X^{1 / 2}\) (c) \(Y=\ln X\)

The random variable \(X\) has the probability distribution $$ f_{X}(x)=\frac{x}{8}, \quad 0 \leq x \leq 4 $$ Determine the probability distribution of \(Y=(X-2)^{2}\).

An aircraft is flying at a constant altitude with velocity magnitude \(r_{1}\) (relative to the air) and angle \(\theta_{1}\) (in a twodimensional coordinate system). The magnitude and direction of the wind are \(r_{2}\) and \(\theta_{2},\) respectively. Suppose that the wind angle is uniformly distributed between 10 and 20 degrees and all other parameters are constant. Determine the probability density function of the magnitude of the resultant vector \(r=\left[r_{1}^{2}+r_{2}^{2}+r_{1} r_{2}\left(\cos \theta_{1}-\cos \theta_{2}\right)\right]^{0.5}\)

Derive the probability density function for a lognormal random variable \(Y\) from the relationship that \(Y=\exp (W)\) for a normal random variable \(W\) with mean \(\theta\) and variance $\omega^{2}.

The computational time of a statistical analysis applied to a data set can sometimes increase with the square of \(N,\) the number of rows of data. Suppose that for a particular algorithm, the computation time is approximately \(T=0.004 N^{2}\) seconds. Although the number of rows is a discrete measurement, assume that the distribution of \(N\) over a number of data sets can be approximated with an exponential distribution with a mean of 10,000 rows. Determine the probability density function and the mean of \(T\).

Power meters enable cyclists to obtain power measurements nearly continuously. The meters also calculate the average power generated over a time interval. Professional riders can generate 6.6 watts per kilogram of body weight for extended periods of time. Some meters calculate a normalized power measurement to adjust for the physiological effort required when the power output changes frequently. Let the random variable \(X\) denote the power output at a measurement time and assume that \(X\) has a lognormal distribution with parameters \(\theta=5.2933\) and \(\omega^{2}=0.00995 .\) The normalized power is computed as the fourth root of the mean of \(Y=X^{4}\). Determine the following: (a) Mean and standard deviation of \(X\) (b) \(f_{Y}(y)\) (c) Mean and variance of \(Y\) (d) Fourth root of the mean of \(Y\) (e) Compare \(\left[E\left(X^{4}\right)\right]^{1 / 4}\) to \(E(X)\) and comment.

A random variable \(X\) has the discrete uniform distribution $$ f(x)=\frac{1}{m}, \quad x=1,2, \ldots, m $$ (a) Show that the moment-generating function is \(M_{X}(t)=\frac{e^{t}\left(1-e^{t m}\right)}{m\left(1-e^{t}\right)}\) (b) Use \(M_{X}(t)\) to find the mean and variance of \(X\).

A random variable \(X\) has the Poisson distribution \(f(x)=\frac{e^{-\lambda} \lambda^{x}}{x !}, \quad x=0,1, \ldots\) (a) Show that the moment-generating function is \(M_{X}(t)=e^{\lambda\left(e^{t}-1\right)}\) (b) Use \(M_{X}(t)\) to find the mean and variance of the Poisson random variable.

The chi-squared random variable with \(k\) degrees of freedom has moment- generating function \(M_{X}(t)=(1-2 t)-k / 2\). Suppose that \(X_{1}\) and \(X_{2}\) are independent chi-squared random variables with \(k_{1}\) and \(k_{2}\) degrees of freedom, respectively. What is the distribution of \(Y=X_{1}+X_{2} ?\)

A continuous random variable \(X\) has the following probability distribution: $$ f(x)=4 x e^{-2 x}, \quad x>0 $$ (a) Find the moment-generating function for \(X\). (b) Find the mean and variance of \(X\).

A random variable \(X\) has the gamma distribution $$ f(x)=\frac{\lambda}{\Gamma(r)}(\lambda x)^{r-1} e^{-\lambda x}, \quad x>0 $$ (a) Show that the moment-generating function of \(X\) is $$ M_{X}(t)=\left(1-\frac{t}{\lambda}\right)^{-r} $$ (b) Find the mean and variance of \(X\).

Let \(X_{1}, X_{2}, \ldots, X_{r}\) be independent exponential random variables with parameter \(\lambda\). (a) Find the moment-generating function of \(Y=X_{1}+X_{2}+\) \(\ldots+X_{r}\) (b) What is the distribution of the random variable \(Y ?\)