(in Supplementary Problem Set 2: Distributions Problem i The Venture Capitalist
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(in Supplementary Problem Set 2: Distributions Problem i The Venture Capitalist (Bernouli & Binomial Distributions) A venture capitalist (VC) decided to set aside $SOM for next year to invest equally among 10 new- venture firms in the out of 20 investments are successful. The others either fail or join the ranks of the "living dead" (companies that are barely surviving and are thus not attractive to other buyers). risky-technology sector that are seeking additional equity capital. Historically, one sts $5M each in 10 new-venture firms using the same selection criteria, then... A. W B. hat is the probability that none of the investments succeed? What is the probability that at least one investment succeeds? at one investment is successful with the above "p (at a minimum), what return receiving the investment capital if the VC expects to would be required from the company double the $5OM invested in the portfollo of companies? D. Ifonly 1 ou t of 20 investments is successful, what i is the minimum number of investments the VC must make to ensure at least 78.5% probability of 1 or more successes? 22 The Project Manager [Triangular Distribution The probability distribution for a technical task's time follows a Beta distribution, and the Project Manager is using triangular distribution as an approximation. The manager's most optimistic time estimate (a) is 2 days, most likely time estimate (c) is 3 days, and most pessimistic time estimate (b) is 7 days based on prior experiences. What is the expected time for this task if it is repeated many times using this approximation? Problem 3 The Car Wash [Exponential Distribution) The owner of a hand car wash for luxury vehicles is considering adjusting his prices for certain types of vehicles in light of rising labor rates and is therefore re-analyzing the probability of a vehicle taking 50% more time than the mean. Time through the car wash follows an exponential distribution with a mean of 10 minutes. What is the probability of a vehicle taking more than 15 minutes? Problem 4 The Architectural Firm [Poisson Distribution) An architectural firm needs at least 5 average-sized jobs per month to cover its fixed expenses, includi salaries. On average, the firm currently receives 8 average-sized jobs per month from its existing clie base. Job arrivals follow a Poisson distribution. What is the probability of the firm meeting this overhead expense? Does the marketing manager need to secure more clients?Explanation / Answer
1)
X= number of investment succeed
X follow binomial with n = 10 , p = 1/20 = 0.05
a) P(X = 0) = (1-p)^10 = 0.95^20
b) P(X >=1) = 1 - P(X =0) = 1 - 0.95^20
c)
Y = return from investments
E(X) = np
E(Y) = M* np
where M is return required for company to double the 50M invested
M*np = 100
M = 100/(np) = 100/(10*0.05) = 200 M
d) let the number of investment= n
P(X>=1) >= 0.785
1 - P(X = 0) >= 0.785
1 - (0.95)^n >=0.785
n >= 29.9672
n =30
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