- What is a normal distribution and why is useful to understand probabilities an

Question

- What is a normal distribution and why is useful to understand probabilities and statistics?. Provide an example of a normal distribution and why can be categorized as normal distribution

- What is the difference between standard normal distribution and normal distribution?

- How we standarize a normal distribution?

- What is a Z-score?, and how we can calculate the z-score for a normal variable?

- How we can use the cummulative probability table for a standard normal distribution to calculate the probability of:

a) less than a certain z-score

b) greater than a z-score

c) between 2 z-scores

Explanation / Answer

All answers, explained in detailes, with formulae wherever required:

1.

Normal distribution is a probability distribution that associates the normal random variable X with a cumulative probability.Its graph depends on two factors - the mean and the standard deviation. The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height of the graph. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow. All normal distributions look like a symmetric, bell-shaped curve.

For example, Probability of getting Heads out of 10000 tosses can be modelled through normal distribution.

becasue we can approximate the distribution into a normal distribution using Mean and deviation.

2. Difference between standard normal distribution and a normal distribution is that standard normal deviation is "normalized" to a mean of 0 and standard deviation of sigma, i.e. all values have been standardized around mean 0 and scaled for deviation. ie.. its symmetricity is around mean of 0 vs. normal distribution where we haven't scaled or moved means.

3. Suppose you have a variable X with params of N(~ Mu, Sigma^2), then standardize this variable as:

Z = X-Mu / Sigma

4. A Z -scores is nothing but the standardized equivalent of X

You can calculate this by using the formula in 3.

6. So, for this go to the Z distribution table , and I will give you an example.

Table here: http://sixsigmastudyguide.com/wp-content/uploads/2014/04/z-table.jpg

Now, open the above link and I will teach you how to calculate the following probabilities:

a.Lets say we want to calculate less than Z = 0.8 = P(Z<c) = P(Z<.8) = .7881

b.Lets say we want to calculate greater than Z = .8 = P(Z>=.8) = 1-P(Z<0.8) = 1-.7881 = .2119
c.Lets say we want to calculate P(.5<Z<.8) = P(Z<.8)-P(Z<.5) = .7881-.5 = .2881

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