Probability and Statistics - Week 2

What does the Central Limit Theorem state?

The Central Limit Theorem in Statistics states that as the sample size increases and its variance is finite, then the distribution of the sample mean approaches normal distribution irrespective of the shape of the population distribution

Central Limit Theorem formula

Z= \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}

Assumptions of Central Limit Theorem

• The drawing of the sample from the population should be random.

• The drawing of the sample should be independent of each other.

• The sample size should not exceed ten percent of the total population when sampling is done without replacement.

• Sample Size should be adequately large.

• CLT only holds for a population with finite variance.

Bayes Theorem formula

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

What is joint probability?

Joint probability is the probability of two or more events occurring simultaneously. It is denoted as P(A and B) and can be calculated by multiplying the probability of each event, taking into account whether the events are independent or dependent. Note: If A and B are dependent, the joint probability is calculated using conditional probability

What is marginal probability?

Marginal probability refers to the probability of an event occurring, irrespective of the outcomes of other variables. It is obtained by summing or integrating the joint probabilities over all possible values of the other variables.
Formula for discrete rv : P(A)=\sum_B P(A,B)

Formula for continuous rv: P(A)=\int_B P(A,B) dB

What is conditional probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. It provides a way to update our predictions or beliefs about the occurrence of an event based on new information.

Formula: P(A|B) = \frac{P(A\cap B)}{P(B)}

Give the definition of Bernoulli distribution

Binomial Distribution is a probability distribution that describes the number of successes in a fixed number of independent binomial trials i.e., each trial can only result in either a success or a failure.

What does the law of large numbers state?

The law of large numbers, in statistics, states that the results of a test on a sample get closer to the average of the whole population as the sample size grows. That is, it becomes more representative of the population as a whole.