Overview(probability distributions)
Whenever we are uncertain about the outcome of any problem, we use the concept of probability and probability distributions. The theory of probability is widely used in finance, healthcare, weather forecast, insurance, sports, etc.
The study of probability distributions is one of the most important parts of statistics.
The concept of probability distribution gives the idea about the probabilities of any random experiment.
The probability distribution indicates the likelihood of an outcome,
Nowadays, statistics and the concept of probability and its distributions has become a very important part of the fields like data science, machine learning, artificial intelligence, computer science etc.
Before moving towards the probability distribution, we will study statistics and probability.
Statistics & probability
Statistics deals with the collection, organizing, analysing, interpreting, and presentation of the data.
Mainly the statistics is divided into two parts-
Descriptive statistics deals with the properties of sample and population and inferential statistics deals, how to take decisions from the data (sample) by using various parameters.
Topics we study in descriptive statistics are- mean, median, mode, SD, variance, skewness, kurtosis etc.
And in inferential statistics we study- regression analysis, analysis of variance which is also known as ANOVA, testing of hypotheses etc.
Probability
We can not predict in many cases what will happen in the future. For example if we toss a coin, we do not know what will be the outcome, whether it is head or tail.
Or if we throw a dice, we can not predict which number will come.
In these cases, we use the idea of probability.
In general, the meaning of probability is possibilities.
It predicts how likely events are to happen. Probability measures the likelihood of an event to occur.
The value of probability always lies between 0 to 1.
If the outcome is impossible then probability is zero, and if the outcome is certain the it is 1.
For example- you all have seen a famous bollywood movie sholay, in which Amitabh bacchan uses a coin, whose coin always comes up head, and we come to know in the climax that the coin he used, had head on both sides.
That means, this is the case of a certain outcome, we know what will come.
The probability of a certain outcome is always 1.
Random Experiment
An experiment in which we know all the possible outcomes but we can not say that or we can not predict which of them will occur when we perform the experiment.
Definition of probability (Classical)
Suppose there are ‘n’ exhaustive cases in a random experiment which is equally likely and mutually exclusive.
Let ‘m’ cases are favourable for the happening of an event A, then the probability of happening event A can be defined as-
Probability of non-happening of the event A is defined as-
The classical definition of probability fails if-
1. The cases are not equally likely
2. The number of exhaustive cases is indefinitely large
Probability distributions
We can write a probability distribution in the form of an equation or a table.
We can calculate the probability distribution from simple events to complex events.
Suppose a coin is tossed. Then its probability distribution can be tabulated as below-
| Number of heads | Probability |
| 0 | 0.25 |
| 1 | 0.50 |
| 2 | 0.25 |
Note- Sum of all the probabilities is always 1.
There are two important terms we use in probability distribution.
One is expected value and the other is variance.
Expected value is the average value of a random variable and variance is the average spread of values around expected values.
There are two type of probability distribution-
1. Discrete probability distributions
2. Continuous probability distributions.
If the random variable is discrete, then it will have a discrete probability distribution.
Examples of discrete probability distributions-
Suppose we throw a dice then the probabilities associated with the outcomes form a discrete uniform distribution, the probabilities of tossing a coin from a Bernoulli distribution and the probabilities of the cars passing through a road within a fixed period from a Poisson distribution.
Following distribution are known as the discrete probability distributions are-
1. Binomial distribution
2. Bernoulli distribution
3. Geometric distribution
4. Hyper-geometric distribution
5. Multinomial distribution
6. Poisson distribution
7. Negative binomial distribution
If the random variable is continuous, it will have a continuous probability distribution.
Following distribution are known as the continuous probability distributions are-
3. Beta distribution
4. Cauchy distribution
5. Gamma distribution
6. Logistic distribution
7. Weibull distribution
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