MATH 407 Final HW Problems
Show that if Z is a normal variable with mean u=0 and standard deviation o=1, then E(Z^3)=0.
Use the integral definition of expected value and the fact that the integrand is odd on a symmetric interval to say it's 0.
A machining operation produces bearing with diameters that are normally distributed with u=3.0005 inches and o=0.0010 inch. Specifications require the bearing diameters to lie in the interval 3.000+/-0.0020 inches. Those outside the interval are considered scrap and must be remachined. With the existing machine setting, what fraction of total production will be scrap?
0.073
If Y has a mgf m(t)=(0.7e^t+0.3)^10, what is P(Y≤5)?
0.150
A random variable Y has a Poisson distribution and is such that p(1)=p(0). What is p(2)?
0.18394
The GPAs of a large population of college students are approximately normally distributed with u=2.4 and o=0.8. What fraction of the students will possess a GPA in excess of 3.00?
0.2266
If students possessing a GPA less than 1.9 are dropping from college, what percent of students will be dropped? (normal with u=2.4, o=0.8)
0.265986
A manufacturing plant uses a specific bulk product. The amount of product used in one day can be modeled by an exponential distribution with B=4 (measurements in tons). Find the probability that the plant will use more than 4 tons on a given day.
0.36788
Of the next 10 earthquakes to strike this region, what is the probability at least one will exceed 5.0 on the Richter scale? (exponential, u=2.4)
0.7355
The percentage of impurities per batch in a chemical product is a random variable Y with density function f(y)= { 12y^2(1-y), 0≤y≤1; 0, elsewhere A batch with more than 40% impurities cannot be sold. a) Integrate the density function directly to determine the probability that a randomly selected batch cannot be sold because of excessive impurities.
0.8208
If Y has a mgf m(t)=e^[6(e^t-1)], what is P(|Y-u|≤2o)?
0.940
The US mint produces dimes with an average diameter of 0.5 inches and standard deviation 0.01. Using Tchebysheff's theorem, find a lower bound for the number of coins in a lot of 400 coins that are expected to have a diameter between 0.48 and 0.52.
300 coins
A soft drink machine can be regulated so that it discharges an average of u ounces per cup. If the ounces of fill are normally distributed with o=0.3 ounce, give the setting for u so that 8-ounce cups will overflow only 1% of the time.
7.31
Use the uniqueness of mgfs to give the distribution of a random variable with mgf m(t)=(0.6e^t+0.4)^3
Binomial, p=0.6, q=0.4, n=3
A circle of radius r has area A=(pi)r^2. If a random circle has a radius that is uniformly distributed on the interval (0,1), what are the mean and variance of the area of the circle?
E(A)=(pi)/3 V(A)=4(pi)^2/45
The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C=100+40Y+3Y^2 relates the cost C of completing this operation to the square of the time to completion. Find the mean and variance of C.
E(C)=1100 V(C)=2,920,000
For a certain section of a pine forest, the number of diseased trees per acre, Y, has a Poisson distribution with mean λ=10. The diseased trees are sprayed with an insecticide at a cost of $3/tree, plus a fixed overhead cost for equipment rental of $50. Letting C denote the total spraying cost for a randomly selected acre, find the expected value and standard deviation for C. What interval would you expect C to lie with a probability of at least 0.75?
E(C)=80 o=sqrt(90) Interval: between 61.026 and 98.974
The weekly amount of downtime Y (in hours) for an industrial machine has approximately a gamma distribution with a=3 and B=2. The loss L (in dollars) to the industrial operation as a result of this downtime is given by L=30Y+2Y^2. Find the expected value and variance of L.
E(L)=276 V(L)=47,664
A food manufacturer uses an extruder that yields revenue for the firm at a rate of $200/hr when in operation. However, the extruder breaks down an average of 2 times every day it operates. If Y denotes the number if breakdowns per day, the daily revenue generated by the machine is R=1600-50Y^2. Find the expected daily revenue for the extruder.
E(R)=1300
For certain ore samples, the proportion Y of impurities per sample is a random variable with density function f(y)={ 3/2 y^2 +y, 0≤y≤1; 0, elsewhere. The dollar value of each sample is W=5-0.5Y. Find the mean and variance of W. (E(Y)=0.7083, V(Y)=0.0483)
E(W)=4.6458 V(W)=0.012075
In the daily production of a certain kind of rope, the number of defects per foot Y is assumed to have a Poisson distribution with mean λ=2. The profit per foot when the rope is sold is given by X, where X=50-2Y-Y^2. Find the expected profit per foot.
E(X)=40
If Y has a density function f(y)={ 3/2 y^2 +y, 0≤y≤1; 0, elsewhere Find the mean and variance of Y.
E(Y)=0.7083 V(Y)=0.0483
Differentiate the mgf m(t)=(pe^t)/(1-qe^t) to find E(Y) and E(Y^2). Then find V(Y).
E(Y)=1/p E(Y^2)=(1+q)/p^2 V(Y)=q/p^2
If Y has distribution function F(y)= {0, y≤0; y/8, 0<y<2; y^2/16, 2≤y<4; 1, y≥4 find the mean and variance of Y.
E(Y)=2.583 V(Y)=1.1597
Find the expected value of the low bids on contracts of the type described there (uniform between 20 and 25).
E(Y)=22.5
Four-week summer rainfall totals in a section of the Midwest US have approximately a gamma distribution with a=1.6 and B=2.0 a) Find the mean and variance of the four-week rainfall totals
E(Y)=3.2 V(Y)=6.4
Find the mean and variance of the percentage of impurities in a randomly selected batch of the chemical. (Beta, a=3, B=2)
E(Y)=3/5 V(Y)=1/25
Use the uniqueness of mgfs to give the distribution of a random variable with mgf m(t)=(0.3e^t)/(1-0.7e^t)
Geometric, p=0.3, q=0.7
Find an interval that would contain L for at least 89% of the weeks that the machine is in use. (E(L)=267, V(L)=47,664)
Interval: between 0 and 934.235
A manufacturer of tires wants to advertise a mileage interval that excludes no more than 10% of the mileage on tires he sells. All he knows is that, for a large number of tires he tested, the mean milage was 25,000 miles, and the standard deviation was 4000 miles. What interval would you suggest?
Interval: between 12,350 and 37,649 miles
Consider the Binomial experiment for n=20 and p=0.05. Use Table 1, Appendix 3, to calculate the binomial probabilities for Y=0,1,2,3,4. Calculate the same probabilities by using the Poisson approximation with λ=np. Compare.
See HW 8 #134 for chart. - Table uses P(Y≤a), but we need P(Y=a), so subtract P(Y≤a-1)
Verify that if Y has a beta distribution with a=B=1, then Y has a uniform distribution over (0,1). That is, the uniform distribution over the interval (0,1) is a special case of a beta distribution.
Show Beta density function with a=B=1 is the same as the density function of uniform distribution on (0,1).
Suppose that Y is a random variable that takes only integer values 1, 2, ... and has distribution function F(y). Show that the probability function p(y)=P(Y=y) is given by p(y)= {F(1), y=1; F(y)-F(y-1), y=2,3,...
Show F(1)=p(1) and F(y)-F(y-1)=p(y)
If Y has a geometric distribution with probability of success p, show that the mgf for Y is m(t)=(pe^t)/(1-qe^t) where q=1-p.
Show it
If Y is a random variable with mgf m(t) and if W is given by W=aY+b, show that the mgf of W is e^bt m(at).
Show it
Use that the mgf of W is e^bt m(at) to prove that, if W=aY+b, then E(W)=aE(Y)+b and V(W)=a^2 V(Y).
Show it, using derivatives
Show that if Y is a random variable with mgf m(t) and U is given by U=aY+b, the mgf of U is e^(bt)m(at). If Y has mean u and variance o^2, use the mgf of U to derive the mean and variance of U.
Show m(t) of U is e^(bt)m(at), E(U)=au+b, and V(U)=a^2o^2
Let Z be a normal variable with parameters u=0 and o=1. Use the density function to show that P(Z<-a)=P(Z>a) whenever a>0. Note that we used this in the notes when applying to symmetry.
Use the integral of the standardized density function and do a variable change to switch the bounds
If Z is a standard normal random variable, find the value z_0 such that a) P(Z>z_0)=0.5 b) P(Z<z_0)=0.8643 c) P(-z_0<Z<z_0)=0.9 d) P(-z_0<Z<z_0)=0.99
a) 0 b) 1.1 c) 1.64 d) 2.57
Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of 7 per hour. During a given hour, what are the probabilities that: a) No more than 3 customers arrive? b) At least 2 customers arrive? c) Exactly 5 customers arrive?
a) 0.08177 b) 0.99270 c) 0.12772
Let Y denote a random variable that has a Poisson distribution with mean λ=2. Find a) P(Y=4) b) P(Y≥4) c) P(Y<4) d) P(Y≥4 | Y≥2)
a) 0.09022 b) 0.14288 c) 0.85712 d) 0.24054
The magnitude of earthquakes recorded in a region of North America can be modeled as having an exponential distribution with mean 2.4, as measured on the Richter scale. Find the probability that an earthquake striking the region will a) exceed 3.0 on the Richter scale b) fall between 2.0 and 3.0 on the Richter scale
a) 0.2865 b) 0.14809
Use Table 4, Appendix 3, to find the following probabilities for a standard normal random variable Z. a) P(0≤Z≤1.2) b) P(-0.9≤Z≤0) c) P(0.3≤Z≤1.56) d) P(-0.2≤Z≤0.2) e) P(-1.56≤Z≤-0.2)
a) 0.3849 b) 0.3159 c) 0.3227 d) 0.1586 e) 0.3613
Let Y be a random variable with mean 11 and variance 9. Use Tchebysheff's Theorem, find: a) A lower bound for P(6<Y<16) b) The value of C such that P(|Y-11|≥C)≤0.09
a) 0.64 b) 10
If Z is a standard normal random variable, what is a) P(Z^2<1)? b) P(Z^2<3.84146)?
a) 0.6826 b) 0.95
A random variable Y has the density function f(y)= { e^y, y<0; 0, elsewhere a) Find E(e^(3y/2)) b) Find the mgf for Y c) Find V(Y)
a) 2/5 b) 1/(1+t) c) 1
Upon studying low bids for shipping contracts, a microcomputer manufacturing company finds that intrastate contracts have low bids that are uniformly distributed between 20 and 25, in units of thousands of dollars. Find the probability that the low bid on the next intrastate shipping contract a) is below $22,000 b) in excess of $24,000
a) 2/5 b) 1/5
Errors in measuring the time of arrival of a wave front from an acoustic source sometimes have an approximate beta distribution. Suppose that these errors, measured in microseconds, have approximately a beta distribution with a=1 and B=2. a) What is the probability that the measurement error in a random selected instance is less than 0.5 u's? b) Give the mean and standard deviation of the measurement errors.
a) 3/4 b) E(Y)=1/3 V(Y)=1/sqrt(18)
If Y has an exponential distribution and P(Y>2)=0.0821, what is a) B=E(Y)? b) P(Y≤1.7)?
a) B=0.8 b) 0.8806
Find the distribution of the random variables that have each of the following mgfs: a) m(t)=(1/3 e^t + 2/3)^5 b) m(t)=e^t/(2-e^t) c) m(t)=e^[2(e^t-1)]
a) Binomial, p=1/3, q=2/3, n=5 b) Geometric, p=1/2, q=1/2 c) Poisson, λ=2
By inspection, give the mean and variance of the random variables associated with the mgfs given in parts a, b, and c. a) m(t)=(1/3 e^t + 2/3)^5 b) m(t)=e^t/(2-e^t) c) m(t)=e^[2(e^t-1)]
a) Binomial: u=1.67, o^2=1.11 b) Geometric: u=2, o^2=2 c) Poisson: u=2, o^2=2
A random variable Y has the following distribution function: F(y)={ 0, y<2 1/8, 2≤y<2.5 3/16, 2.5≤y<3 1/2, 4≤y<5.5 5/8, 5.5≤y<6 11/16, 6≤y<7 1, y≥7 a) Is Y a continuous os discrete random variable? Why?
a) Discrete, finite range, distribution is a step function, not continuous
Let m(t)=1/6 e^t + 2/6 e^2t + 3/6 e^3t. Find the following: a) E(Y) b) V(Y) c) The distribution of Y
a) E(Y)=7/3 b) V(Y)=5/9 c) p(1)=1/6, p(2)=2/6, p(3)=3/6
Let Y be a random variable with p(y) given below: p(1)=0.4, p(2)=0.3, p(3)=0.2, p(4)=0.1 a) Give the distribution function F(y) b) Sketch the distribution function
a) F(0)= { 0, y<1 0.4, 1≤y<2 0.7, 2≤y<3 0.9, 3≤y<4 1, y≥4 b) Sketch
Suppose that Y has a uniform distribution over the interval (0,1). a) Find F(y) b) Show that P(a≤Y≤a+b), for a≥0, b≥0, and a+b≤1 depends only upon the value of b.
a) F(y)= { 0, y<0; y, 0≤y≤1; 1, y>1 b) Show P(a≤Y≤a+b) = b
Consider a random variable with a geometric distribution; that is p(y)=q^y p, y=1,2,3..., 0<p<1 a) Show that Y has a distribution function F(y) such that F(i)=1-q^i, i=0,1,2,... and that in general, F(y)={ 0, y<0; 1-q^i, i≤y<i+1 b) Show that the preceding cumulative distribution function has the properties given in Theorem 4.1.
a) Show F(i)=1-q^i b) Limit to -inf=0; Limit to inf=1; as y increases, p(y) increases: i≤i+1 and q≤1
As a measure of intelligence, mice are timed when going through a maze to reach a reward of food. The time (in seconds) required for any mouse is a random variable Y with a density function given by: f(y)={ b/y^2, y≥b; o, elsewhere a) Show that f(y) has the properties of a density function b) Find F(y). c) Find P(Y>b+c) d) If c and d are both positive constants such that d>c, find P(Y>b+d | Y>b+c)
a) Show that f(y)≥0 and integral over all is 1 b) F(y)={ -b/y +1, y≥b; 0, elsewhere c) b/(b+c) d) (b+c)/(b+d)
If Y is a continuous random variable with mean u and variance o^2 and a and b are constants, use Theorem 4.5 to prove the following: a) E(aY+b)=aE(Y)+b=au+b b) V(aY+b)=a^2V(Y)=a^2o^2
a) Show using integral b) Show using definition
A Bernoulli random variable is one that assumes only two values, 0 and 1 with p(1)=p and p(0)=1-p=q. a) Sketch the corresponding distribution function b) Show that this distribution has the properties given in Theorem 4.1.
a) Sketch b) Limit to -inf=0; Limit to inf=1; as y increases, p(y) increases: 0≤1 and q≤1
Suppose that y has a gamma distribution w parameters a and B. a) If c is any positive or negative value such that a+c>0, show that E(Y^c)=(B^cG(a+c))/G(a) b) Why did your answer in part a require that a+c>0? c) Show that, with c=1, the result in part a gives E(Y)=aB d) Use the result in part a to give an expression for E(sqrt(Y)). What do you need to assume about a? e) Use the result in part a to give an expression for E(1/Y), E(1/sqrt(Y)), and E(1/Y^2). What do you need to assume about a in each case?
a) Use integral definition of expected value to show it b) G(x) is not defined when x≤0 c) Show it d) E(sqrt(Y))=(B^1/2 G(a+1/2))/G(a), assume a>0 e) E(1/Y)=(B^-1 G(a-1))/G(a), assume a>1 E(1/sqrt(Y))=G(a-1/2)/(B^1/2 G(a)), assume a>1/2 E(1/Y^2)=1/(B^2(a-1)(a-2)), assume a>2
The proportion of time per day that all checkout counters in a supermarket are busy is a random variable Y with density function: f(y)={ cy^2(1-y)^4, 0≤y≤1; 0, elsewhere a) Find the value of c that makes f(y) a probability density function. b) Find E(Y).
a) c=105 b) 0.375
The length of time required by students to complete a one-hour exam is a random variable with a density function given by: f(y)={ cy^2 +y, 0≤y≤1; 0, elsewhere a) Find c b) Find F(y) c) Graph f(y) and F(y) d) Use F(y) in part b to find F(-1), F(0), and F(1). e) Find the probability that a randomly selected students will finish in less than half an hour. f) Given that a particular student needs at least 15 minutes to complete the exam, find the probability that she will require at least 30 minutes to finish.
a) c=3/2 b) F(y)= { 0, y<0; 1/2 (y^3 +y^2), 0≤y≤1; 1, y>1 c) Graph d) F(-1)=0, F(0)=0, F(1)=1 e) 0.1875 f) 0.8455
Let the distribution function of a random variable Y be F(y)= {0, y≤0; y/8, 0<y<2; y^2/16, 2≤y<4; 1, y≥4 a) Find the density function of Y. b) Find P(1≤Y≤3) c) FindP(Y≥1.5) d) Find P(Y≥1 | Y≤3)
a) f(y)= {0, y≤0; 1/8, 0<y<2; y/8, 2≤y<4; 0, y≥4 b) 0.4375 c) 0.8125 d) 0.7778
Identify the distributions of the random variables with the following mgfs: a) m(t)=(1-4t)^-2 b) m(t)=1/(1-3.2t) c) m(t)=e^(-5t+6t^2)
a) gamma, a=2, B=4 b) exponential, B=3.2 c) normal, u=-5, o^2=12
Suppose that Y has a density function f(y)= { ky(1-y), 0≤y≤1; 0, elsewhere a) Find the values of k that makes f(y) a probability density function. b) Find P(0.4≤Y≤1) c) Find P(0.4≤Y<1) d) Find P(Y≤0.4 | Y≤0.8) e) Find P(Y<0.4 | Y<0.8)
a) k=6 b) 0.648 c) 0.648 d) 0.3929 e) 0.3929
Suppose Y has a random variable with mgf m(t). a) What is m(0)? b) If W=3Y, show that the mgf of W is m(3t). c) If X=Y-2, show that the mgf of X is e^-2t m(t).
a) m(0)=1 b) Show it c) Show it
Example 4.16 derives the mgf for Y-u, where Y is normally distributed with mean u and variance o^2. a) Use the result from Example 4.16 and m(t) of aY+b =e^(bt)m(at) to find the mgf for Y b) Differentiate the mgf found in part a to show that E(Y)=u and V(Y)=o^2.
a) m(t) of Y =e^(ut+(1/2)t^2o^2) b) Differentiate to show E(Y)=u and V(Y)=o^2
Suppose that the waiting time for the first customer to enter a retail shop after 9pm is a random variable Y with an exponential density function given by f(y)= { (1/θ)e^(-y/θ), y>0; 0, elsewhere a) Find the mgf for Y. b) Use the answer from part a to find E(Y) and V(Y)
a) m(t)=1/(1-θt) b) E(Y)=θ V(Y)=θ^2
Would you expect C to exceed 2000 very often? (E(C)=1100, V(C)=2,920,000)
k=0.53, so likely C exceeds 2000 often
A machine used to fill cereal boxes dispenses, on the average, u ounces per box. The manufacturer wants the actual ounces dispensed Y to be within 1 ounce of u at least 75% of the time. What is the largest value of o, the standard deviation of Y, that can be tolerated if the manufacturer's objectives are to be met?
o=1/2