SOLUTION: PROBABILITY DISTRIBUTIONS (Binomial and Poisson)
ABC company estimates the net profit on a new product ,that it is launching, to be Rs. 30,00,000 if it is successful, Rs. 10,00,
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Question 874804: PROBABILITY DISTRIBUTIONS (Binomial and Poisson)
ABC company estimates the net profit on a new product ,that it is launching, to be Rs. 30,00,000 if it is successful, Rs. 10,00,000 if it is moderately successful and a loss of Rs. 10,00,000 if it is unsuccessful. The firm assigns the following probabilities to the different possibilities: Successful 0.15, moderately successful 0.25 and unsuccessful 0.60. Find the expected value and variance of the net profits.
A survey conducted over last 25 years indicated that in 10 years the winter was mild, in 8 years it was cold and in the remaining 7 years it was very cold . A company sells 1,000 woolen coats in the mild cold year, 1,300 in a cold year and 2000 in a very cold year. You are required to find the yearly expected profit of the company if a woolen coat costs Rs. 173 and is sold to stores for Rs. 248,
A company introduces a new product in the market and expects to make a profit of Rs. 2.5 lacs during first year if the demand is ‘good’, Rs. 1.5 lacs if the demand is ‘moderate’ and a loss of Rs 1 lac if the demand is ‘poor’ . Market research studies indicate that the probabilities for the demand to be good and moderate are 0.2 and 0.5 respectively. Find the company’s expected profit and standard deviation.
Find the mean and variance of the following probability distribution:
X = x 0 1 2 3 4 5
P(X = x) 3α 4α 4α 2α 2α 1α
Consider a random variable with the following probability distributions: P(x=0) = 0.1, P(x=1) = 0.2,P(x=2)=0.3, Px=3)=0.3 and P(x=4) =0.1
Find P(x≤2)
Find P(1
Find P(x≥0)
Find the expected value of X
Find the standard deviation of x.
The following examples are experiments and their associated random variables . In each case identify the values the random variable can take on and state whether the random variable is discrete or continuous.
Experiment Random variable
Take a 20 question examination Number of questions answered correctly
Observe cars arriving at a tollbooth
For 1 hour Number of cars arriving at the tollbooth
Audit 50 tax returns Number of returns containing errors
Observe an employee’s work
For 8 hours Number of nonproductive hours
Weigh a shipment of goods Number of pounds
Build a new Library Percentage of project completed after 6 months
Data were collected on the number of operating rooms in use at Tampa General Hospital over a 20-day period. On 3 of the days only 1 operating room was use; on 5 days ,2 were use; on 8 days, 3 were used; and on 4 days all 4 rooms were used.
Use the relative frequency approach to construct a probability distribution for the number of operating rooms in use on any given day.
Draw a graph of probability distribution
Find the expectation and standard deviation.
Consider a binomial experiment with 2 trials and p = 0.4
Compute the probability of 1 success
Compute f(0)
Compute f(2)
Find the probability of at least 1 success
Find the expected value, variance, and standard deviation
When a new machine is functioning properly, only 3% of the items produced are defective. Assume that we will randomly select two parts produced on the machine and that we are interested in the number of defective parts found.
Describe the conditions under which this situation would b e a binomial experiment.
How many experimental outcomes yield 1 defect?
Compute the probabilities associated with finding no defects, 1 defect, and 2 defects.
Military radar and missile detection systems are designed to warn a country of enemy attacks. A reliability question deals with the ability of the detection system to identify an attack and issue the warning. Assume that a particular detection system has a 0.90 probability of detection a missile attack. Answer the following questions .
What is the probability that 1 detection system will detect an attack?
If 2 detection systems are installed in the same area and operate independently, what is the probability that at least 1 one of the systems will detect the attack?
If 3 systems are installed, what is the probability that at least one of the systems will detect the attack?
Would you recommend that multiple detection systems be operated? Explain.
If a production unit is made up from 20 identical components and each component has a probability of 0.25 of being effective, what is the average number of defective components in a unit? Further, What is the probability that in a unit (i) less than 3 components are defective?(ii)exactly 3 components are defective.
There are 24 battery cells in a box containing 6 defective cells that are randomly mixed. A customer buys 3 cells. What is the probability that he gets one defective cell?
The average number of customer arrivals per minute at a super bazaar is 2. Find the probability that during one particular minute (i)exactly 3 customers will arrive, (ii)at the most two customers will arrive,(iii)at least one customer will arrive.
A car hire firm has two cars which it hire out every day. The number of demands for a car on each days is distributed as a Poisson variate with mean 1.5. Calculate the proportion of days on which neither car is used and proportion of days on which some demand is refused.
An executive makes, on an average, 5 telephone calls per hour at a cost which may be taken as Rs2 per call, determine the probability that in any hour the telephone calls’ cost (i) exceeds Rs 6,(ii)remains less than Rs.10
The number of accidents in a year attributed to taxi drivers in a city follows Poisson distribution with mean 3. Out of 1,000 taxi drivers, find approximately the number of drivers with (i)no accident in a year, (ii) more than 3 accidents in a year.
Suppose that we are concerned with the occurrence e of major defects in a section of highway one month after resurfacing. We assume that the probability of a defect is the same for any two intervals of equal length, and that the occurrence or nonoccurrence of a defect in any one interval is independent of the occurrence or nonoccurrence in any other interval. Suppose that major defect s occur at the average rate of two per mile. Find the probability that
No major defects will occur in a particular 3-mile section of the highway.
At least two major defects in a 3-mile section of the highway.
For a Binomial distribution mean = 12, variance = 10. Find the parameters of the distribution , Also find the probability that the variate takes non-zero value.
For a Binomial variate with parameters n = 14 & p = 0.4, write down the p.m.f. , mean and S.D.
In a throw of a die 5 or 6 is considered a success. Find the mean and variance of the number of successes and compute P ( x ≥ 3),if a die is thrown 6 times.
If X follows binomial distribution with parameters 10 and 0.6. find (i) E(X-6) ii)E(X-6)/10
iii) E(X-6)2
On an average 1 in every 50 valves manufactured by a firm is substandard. If the valves are supplied in packets of 20 each, find the probability that the packets will contains at least one substandard valve.
In a poisson distribution P2) =4 P(3), find P(4), P(x<3) & moment coefficient of skewness and kurtosis.
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