## Exponential Distribution

- The time between events in Poisson process (i.e. inverse of Poisson)
- Poisson: Events per single unit of time
- Exponential: Time per single events
- Pre-req’s:
- Events must occur at constant rate
- Events must be independent of each other

- Exponential is a continuous distribution as it measure time which can be fractioned or divided
- Since it is continuous distribution, we will have probability distribution function(pdf)
- In Poisson distribution

```
mean = λ
```

- In exponential distribution

```
mean = μ = 1/λ
```

- Example:
- Unique visitors arrive at the quality thought recpetion by a Poisson process at an average of 3 per hour.
- Find the probability that the next visitor arrives
- with in 10 minutes
- after 30 minutes
- in exactly 25 minutes => 0

- Excel Sheet
- EXPON.DIST(x,lambda,cumulative)

## CHI-Squared distribution -> 𝛘2 distibution

- IF Z1,Z2……Zk are the independent samples from standard Normal distribution
- Chi-squared distribution is primarily used in hypothesis test for comparing Goodness of fit (theoreticial probabiliites vs Outcome Probabilities)
- Expected Value and variance
- Visualization:
- TEST Statistic
- Example:
- In an engineering class of 75 students, 11 are left-handed. Does this class fit the prevailing theory that 12% of the people are left handed
- CONCLUSION:
- There is not enough evidence to suggest the proportion of left handedness is other than 0.12

- A poker dealing machine is supposed to deal cards at random from an infinite deck
- In a test, you counted 1600 cards and observed the following
- Spades: 404
- Hearts: 420
- Diamonds: 400
- Clubs: 376

- Is is set equally likely

- In a test, you counted 1600 cards and observed the following

- Refer Here for the excel sheet containing chi-squared tests