3. Standard normal distribution(Z- distribution)
• Let the random variable X has standard normal or Gaussian distribution with mean 𝜇 and standard
deviation 𝜎 ,
• Then the random variable Z is said to have a standard normal distribution if 𝑍 =
𝑋 −𝜇
𝜎
,
• the pdf of standard normal distribution is given by:
In other words, a normal distribution with mean zero (µ=0) and variance one (σ2=σ=1) is
called the standard normal distribution (Z- distribution).
Note that Z ~ N(0, 1)
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7. Application of normal distribution
Normal distribution play very important role in statistical theory because of the following reasons
i) Most of the distributions occurring in the practice e.g. Binomial, Poisson, Hyper geometric
distributions, etc., can be approximated by normal distribution
ii) Most of the sampling distributions e.g. Student t, F-distribution, Chi-square distributions etc., tend
to normality for large samples.
iii) Even if a variable is not normally distributed, it can sometimes be brought to normal form by
simple transformation of variable. For example, if the distribution of X is skewed, the distribution
of square root of X might come out to be normal.
iv) Many of the distributions of sample statistics e.g. the distribution of sample mean, sample variance
etc., tend to normality for large samples.
v) Normal distribution finds large applications in Statistical Quality Control in industry for setting
control limit.
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10.
11. Applications of the Exponential Distribution:
1. Time between telephone calls
2. Time between machine breakdowns
3. Time between successive job arrivals at a computing center
