Probability  Part 2  Probability Distributions  A Tutorial with examples and solved problemsIn this tutorial, we shall discuss certain discrete and continuous probability distributions: Discrete distributions:
Continuous distributions:
Q: Let X ~ N(μ, σ2). Find the moment generating function of X. Q: Suppose that in a class 5% girls are shorter than 5ft. How many girls should be present in each class in order that the probability of atleast one girl shorter than 5ft is 50% or more? Q: The napkins produced by a company are defective with probability 0.01, and are independent of each other. The company sells the napkins in packs of size 10 and offers a moneyback guarantee if more than one of the 10 napkins in the pack is found to be defective. If you buy 3 packs, what is the probability that at most one pack will be returned? Q: A binomial variable on 80 trials has 6 as its std deviation. Is this statement valid? Q: A machine produces 5% defective items. 12 items are inspected every hour. The machine has to be shut down if more than 2 defectives are found. What is the probability that the machine has to be shut for the first time at the time of the 3rd inspection? Q: Ayush is given questions in a Mathematics exam one by one until he fails to answer correctly. The probability that Ayush answers a question correctly is m. The probability that he will quit after answering an odd number of questions is 0.9. Find the value of m. Q: If the probability of a defective bulb is 0.1; find the moment of skewness and kurtosis for 400 bulbs following Binomial distribution. Q: A device is to be packed with 3 parts in each box. Each part is checked before packing and the probability that the part meets the specification is 0.9. X denotes the number of parts that must be checked to fill the box. Find out the pmf, E(X) and V(X). Q: A manufacturer finds that 5% of his products are defective. If he sells products in boxes of 200 and guarantees not more than 0.5% of the products will be defective, what is the probability that the box will fail to meet guarantee? Q: In Belgaon, internet service is available at an internet cafe from 10am to 6pm every day. No. of people browsing the internet follow a Poisson distribution with mean = 50. For a given day, find the probability that (i) the first user will after at 10:15. (ii) The first user will arrive between 10:30 and 11 am. Q: The average number of people watching movies during 9:00a.m. to 9:00 p.m. is thrice the average number during 9:00 p.m. to 9:00 a.m. at a traffic junction. Given no people watch a movie during a day (24 hour period), the conditional probability that the show time was between 9:00 p.m. to 9:00 a.m. is thrice the conditional probability that the show time was between 9:00 a.m. to 9:00 p.m. What is average number of people watching movies during the day (24 hour period)? Q: In a magazine of 60 pages, there are 6 typing errors. Assuming Poisson’s distribution for number of errors per page, find the probability that a randomly chosen 3 pages will contain 0 errors. Q: The life of a component in a machine are independently distributed with the kth component life having an exponential distribution with mean = 2k. What is the probability that exactly two systems are working at the end of 8 years? (k = 1,2,3) Q: The continuous rv X is uniformly distributed with mean 1 and variance 3. Find P(X<0). The devices produced by producer ‘A’ follow a Gamma distribution and have mean 4 and variance 8 whereas those produced by ‘B’ follow exponential distribution with mean 2 and variance 4. An organisation procures 75 units from A. A unit selected at random is found to be working after 12 years. Then find the probability that the device was produced in A. Q: The time for orders of shirts from a manufacturer is known to have a gamma distribution with a mean of 20 days and a standard deviation of 10 days. Determine the probability of receiving an order within 15 days of placement date. Q: The life span of a part of a mechanical device follows a Gamma distribution. The average life and most likely life of the component are 24000 days and 22000 days respectively. Determine variance of the life span of the part. Q: Find the median of a normally distributed rv with usual parameters. Q: The probability that a patient recovers from a disease is 0.4. If 1000 people are known to be suffering from this disease. What is the probability that less than 30 recover? Q: Assume that the time required by a swimmer to complete 1 lap is a normal rv with mean 10 sec and standard deviation 0.25 sec. What is is the shortest interval of time that has a probability of 0.95? Q: The time required by weavers to complete a small portion of a Kashmiri carpet is a normal rv with mean 4min 1sec and standard devation 2 sec. What is the probability that the weaver completes his portion in i. less than 4mins? ii. More than 3min 55secs? Q: Some burglaries occur in Kalyani Nagar in a Poisson process with mean 0.5. Find the probability that i. no more than 10 burglaries occur in a month. ii. No less than 17 burglaries in a month. Q: Let Y denote the radius in cm. of certain type of disc. Assume that Y has a lognormal distribution with parameters μ = 0.8 and σ = 0.1. Find (i) the probability that a randomly selected disc has radius more than 2.7 mm. (ii) Between what two values will Y fall with probability 0.95? Q: The price X of a quantity follows a lognormal distribution with mean 7.43 and variance 0.56. Find mean and variation of Y if X=e^{Y}.
