Question

 

1. Define the characteristics of a probability measure, and explain the difference between a theoretical probability distribution and an empirical probability distribution (a priori vs. a posteriori). (1 point)
2. Describe the characteristics of a normal distribution. (1 point)
3. Explain the importance of the central limit theorem. (1 point)

Answer 1

CHARACTERISTICS OF PROBABILITY MEASURE:

Suppose   (Ω, F, P) is a probability space and probability measure is a function on (Ω, F) as P : F → [0, 1] .

1. P(Ω) = 1

i.e. Probability of Ω is one and

2. If F₁, F₂,​​​​​​​ F₃​​​​​​​, F₄​​​​​​​, ................. ∈ F is a collection of disjoint members in F, then

i.e. Probability of union of all A_i' s is equal to summation of their probabilities .

DIFFERENCE BETWEEN THEORETICAL PROBABILITY DISTRIBUTION and AN EMPIRICAL PROBABILITY DISTRIBUTION

The differences between these two can be stated as follows :

Theoretical Probability	Empirical Probability
1.The theoretical probability of an event is an expected probability based upon knowledge of the experiment under study. 2. Theoretical Probability is the priori probability	1.The empirical probability or the experimental probability is an estimate that is calculated from the observations collected during the experiment . 2.Empirical Probability is the posteriori probability

CHARACTERISTICS OF A NORMAL DISTRIBUTION :

1. Normal Distribution is symmetric around its mean.

2. Normal Distribution is unimodal.

3 . Normal Distribution is asymptotic. .

4 . Mean , Median and Mode of Normal Distribution is equal.

IMPORTANCE OF CENTRAL LIMIT THEOREM :

Central limit theorem is important because we get the results and the distribution of the original population does not matter as the sampling distribution approaches normality in Central limit theorem. Hence it is significantly important in statistics .