Chapter 5- Binomial and Poisson distribution
Medical billing errors and fraud are on the rise. According to the MBAA website, 8 out of 10 times, the medical bills that you get are not right. If a sample of 10 medical bills is selected, what is the probability that: (a) 0 medical bills will contain errors? (b) exactly 5 medical bills will contain errors? (c) more than 5 medical bills will contain errors? (d) What are the mean and standard deviation of the probability distribution?
(a) 0.0000001024 (b) 0.0264 (c) 0.9672 (d) mean=8 standard deviation=1.2650
The probability that a smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8. You have 2 such alarms in your home and they operate independently. (a) What is the probability that both sound an alarm in the presence of smoke? (b) What is the probability that neither sound an alarm in the presence of smoke? (c) What is the probability that at least one sounds an alarm in the presence of smoke?
(a) 0.64 (b) 0.04 (c) 0.96
Subscribers to Investment Advice White Letters perform security transactions at the rate of five trades per month. Assume that one of the subscribers performs transactions at this rate and the probability of a transaction for any two months is the same and the number of transactions in one month is independent of the number of transactions in another month. (a) What is the mean number of transactions per month for this subscriber? (b) What is the variance of the number of transactions per month for this subscriber? (c) What is the probability that exactly ten security transactions will be conducted in one month? (d) What is the probability that at least five security transactions will be conducted in one month? (e) What is the probability that no more than five security transactions will be conducted in one month?
(a) 5 (b) 5 (c) 0.018 (d) 0.560 (e) 0.616
The United Out Corps Reports blog notes that the National Insurance Crime Bureau says that Miami-Dade, Broward, and Palm Beach counties account for a substantial number of questionable insurance claims referred to investigators. Assume that the number of questionable insurance claims referred to investigators by Miami-Dade, Broward, and Palm Beach counties is distributed as a Poisson random variable with a mean of 7 per day. (a) What assumptions need to be made so that the number of questionable insurance claims referred to investigators by Miami-Dade, Broward, and Palm Beach counties is distributed as a Poisson random variable? Making the assumptions in (a), what is the probability that: (b) 5 questionable insurance claims will be referred to investigators by Miami-Dade, Broward, and Palm Beach counties in a day? (c) 10 or fewer questionable insurance claims will be referred to investigators by Miami-Dade, Broward, and Palm Beach counties in a day? (d) 11 or more questionable insurance claims will be referred to investigators by Miami-Dade, Broward, and Palm Beach counties in a day?
(a) There is a set mean of questionable insurance claims. (b) 0.1277 (c) 1.5708 (d)-0.61597
The average number of customers who enter DMV in an hour is 32. Find the probability that 25 customers enter in an hour.
=(e^-32)(32^25)/25!=0.035
What is an area of opportunity?
A continuous unit or interval of time, volume, or such area in which more than one occurrence of an event can occur.
What is a continuous variable?
A continuous variable produces outcomes that come from a measurement.
What is a discrete variable?
A discrete variable produces outcomes that come from a counting process.
What is binomial distribution?
A fixed number of observations, n. Each observation is categorized as to whether or not the "event of interest" occurred. Constant probability for the event of interest occurring (pi) for each observation.
What is a probability distribution for a discrete variable?
It is a mutually exclusive listing of all possible numerical outcomes for that variable and a probability of occurrence associated with each outcome.
What is the Poisson distribution formula?
P(X=x|lambda)= (e^-lambda)(lambda^x)/x! x= number of events in an area of opportunity lambda= expected number of events e= base of the natural logarithm system
What is the binomial distribution formula?
P(X=x|n, pi)=(n!/x!(n-x)!)(pi^x)(1-pi)^n-x where: x= number of "events of interest" n= sample size pi= probability of "event of interest"
What is the Poisson distribution?
The Poisson distribution is used when one is interested in the number of times an event occurs in a given area of opportunity.
What is another term for "expected value"?
The mean
What is the formula to find the standard deviation of the binomial distribution?
o= square root of (n)(pi)(1-pi)
What is the formula for the standard deviation of the Poisson distribution?
o= square root of lambda
What is the formula for the variance of a discrete variable?
o^2=(xi-E(X))^2P(X=xi)
What is the formula for the standard deviation of a discrete variable?
take the variance formula and take the square root
What is the formula to find the mean of the binomial distribution?
u= (n)(pi)
What is the formula to find the expected value?
u=E(X)=xiP(X=xi)
What is the formula for the mean of the Poisson distribution?
u=lambda
What is the probability of one success in five observations if the probability of an event of interest is 0.1?
x=1, n=5, and pi=0.1 = (5!/1!(5-1)!)(0.1)^1(1-0.1)^5-1 =(5)(0.1)(0.9)^4 =0.32805
Suppose the probability of purchasing a defective computer is 0.02? What is the probability of purchasing 2 defective computers in a group of 10?
x=2, n=10, pi=0.02 =(10!/2!(10-2)!)(0.02)^2(1-0.02)^10-2 =(45)(0.0004)(0.8508) =0.01531
