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### Probability Distributions- Discrete and Continuous- with examples- Bernouilli/Binomial/Geometric/Uniform/Exponential/Gamma, etc.

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## Probability - Part 2 - Probability Distributions - A Tutorial with examples and solved problems

In this tutorial, we shall discuss certain discrete and continuous probability distributions:-

## Discrete distributions:

1. Bernoulli trials
2. Binomial distributions
3. Geometric distributions
4. Negative binomial
5. Hypergeometric distributions
6. Poisson distribution

## Continuous distributions:

1. Uniform distributions
2. Exponential distributions
3. Gamma distributions
4. Normal distributions
5. Lognormal distributions

## Here are some of the problems solved in this tutorial :

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 money-back 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 log-normal 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=eY.

## Complete Tutorial with Examples, Plots, Problems and Solutions :

### In case you're interested, here are the tutorials we have, related to probability and probabilistic distributions :

 Probability - Part Zero - A Very Basic Introduction Probability - Part 1 - Basic Probability Definitions, Random Variables Probability - Part 2 - A Tutorial on Probability Distributions Probability - Part 3 - Joint Probability, Bivariate Normal Distributions, Functions of Random Variable,Transformation of Random Vectors