Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Normal Distribution Normal Distribution: Suppose 𝑋 is the set of continuous set of data with mean 𝜇 and variance 𝜎 2 and the graph of p.d.f of this set of data is bell shape then X is said to be normally distributed. 𝜇 and 𝜎 2 are said to be parameter of 𝑋. If X is normally distributed then it can be written as 𝑋~(𝜇, 𝜎 2 ) Examples. 𝑋~(3, 0.9) 𝑋~(14, 4) 𝑋~(78, 25) Curve of normal distribution: The normal distribution curve has the following properties. It is bell shaped. It is symmetrical about mean. It extends from − ∞ 𝑡𝑜 + ∞. The total area under the curve is 1. Approximately 99.8% of the distribution lies within 3 standard deviation of mean. Standard normal variable: If a normal has mean 0 and variance 1 then it is called standardized normal distribution and it is denoted by Z. 𝑍~(0, 1) Probability density function of standard normal distribution: The p.d.f of standard normal distribution is denoted by ∅(𝑧) and it gives the area under the curve before the value of 𝑧. Use of standard normal table: To find the value of ∅(𝑧) at given value of z the number must be in three decimal places. First decimal place is looked in first column and second decimal place is looked in first row. Find the crossed value. Third decimal place is looked in “Column of ADD”. This value is added into last two digits of crossed value. Q.1 Find the value of function ∅(𝑧), i. ∅(1.234) iii. ∅(1.6789) ii. ∅(2.5) iv. ∅(2.45) Relation between area under the graph and probability of standardized normal Variable: Basic Rules: P(Z < 𝑎) = ∅(𝑧) P(Z > 𝑎) = P(Z < −a) ∅(−𝑎) = 𝟏 − ∅(𝑎) P(a < Z < b) = ∅(𝑏) − ∅(𝑎) P(|Z| < 𝑎) = P(−a < Z < a) P(|Z| > 𝑎) = P(Z < −𝑎 𝑜𝑟 Z > 𝑎) = 𝑃(𝑍 < −𝑎) + 𝑃(𝑍 > 𝑎) Q.2 If 𝑍~(0, 1), find i. P(Z < 0.16) ii. P(Z < −1.346) iii. P(Z > 2.9) iv. P(Z < 0.1362) v. P(|Z| < 1.12) vi. P(|Z| > 1.89) Use of standard normal table to find ∅−𝟏 (𝑧): Look the probability in main body of the table. If is not appear then look the value is less than it. Find in which row and column it is lying. Column and row will make first and second decimal places of given value. Compiled By : Sir Rashid Qureshi www.levels.org.pk