Wk3 DQ1| Discussion Question 1 |Checked by Turnitin
What are the differences between Binomial Distribution and Poison Distribution?
A recent CBS News survey reported that 67% of adults felt the U.S. Treasury should continue making pennies. Suppose we select a sample of 15 adults.
a. How many of the 15 would we expect to indicate that the Treasury should continue making pennies? What is the standard deviation?
b. What is the likelihood that exactly eight adults would indicate the Treasury should continue making pennies?
c. What is the likelihood at least eight adults would indicate the Treasury should continue making pennies?
Ans:
In the third week of our course, we discussed the various concepts of probability which included probability distribution as well. A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range (Kenton, 2018). This week 3 DQ 1 is mainly focused on the types of probability distribution (Binomial Distribution and Poison Distribution) along with a numerical analysis.
Difference between Binomial Distribution and Poison Distribution
Binomial distribution is defined as one of the types of probability distribution in which the probability of repeated number of trials is studied. Poisson Distribution is defined as one of the types of probability distribution which gives the count of independent events occurs randomly with a given period of time (S, 2016).
Basis | Binomial Distribution | Poison Distribution |
Definition | Binomial distribution is defined as one of the types of probability distribution in which the probability of repeated number of trials is studied. | Poisson Distribution is defined as one of the types of probability distribution which gives the count of independent events occurs randomly with a given period of time. |
Nature | It is Biparametric in nature which indicates that it is featured by two parameters ‘n’ and ‘p’. | It is Uniparametric in nature which indicates that it is featured by only one parameter ‘m’. |
Outcomes | In case of binomial distribution, there are only two possible outcomes (success or failure). | In case of poison distribution, there are unlimited numbers of possible outcomes. |
Mean and Variance | In case of binomial distribution, Mean is greater than Variance. | In case of poison distribution, Mean is equal to Variance. |
Business Applications | Binomial Distribution can be used by Banks and other financial institutions to determine the likelihood of borrowers defaulting and figuring out how much money to keep in reserve, or how much to loan.
| Poison Distribution can be used by service industries to predict customer sales on particular days of the year as well as can be useful in estimating demand and supply. |
Example | Tossing a Coin (Outcomes: Head or Tail) | Number of people using ATM located outside the XYZ office.(Outcomes: Unlimited) |
Formula | P (X) = ^{n}C_{x} p^{x} q^{n-x} | P(X) = µ^{x }e^{– µ}/x! |
Numerical Solution:
Given Information;
- Number of trails (n) =15
- Probability of Success (p) = 0.67
- Probability of failure (q) = 1-p = 1- 0.67 = 0.33
Numerical Answer (A):
- Expected Value = 15*0.67 = 10.05(approx. 10)
Therefore, the expected value is 10 which indicate that the Treasury should continue making pennies.
- Standard deviation (σ) = √nπ (1−π) = √15×0.67(1−0.67) =1.82
Hence, the standard deviation is 1.821.
Numerical Answer (B):
As per the binomial distribution;
- P(x = 8) = ^{n}C_{x .}p^{x}q^{n-x}= ^{15}C_{8}*0.67^8*0.33^^{15-8}
=0.1113
Therefore, the likelihood that exactly eight adults would indicate the Treasury should continue making pennies is 0.1113.
Numerical Answer (C):
- P (X≥8) = 1−P(X<8) =1−P (X≤7) =1-0.0837 = 0.91629
Therefore, the likelihood that at least eight adults would indicate the Treasury should continue making pennies is 0.91629.
References
Kenton, W. (2018, July 6). Probability Distribution. Retrieved from Investopedia: https://www.investopedia.com/terms/p/probabilitydistribution.asp
S, S. (2016, May 10). Difference Between Binomial and Poisson Distribution. Retrieved from Key Differences: https://keydifferences.com/difference-between-binomial-and-poisson-distribution.html