This GSEB Class 12 Commerce Statistics Notes Part 2 Chapter 3 Normal Distribution Posting covers all the important topics and concepts as mentioned in the chapter.
Normal Distribution Class 12 GSEB Notes
Normal Distribution:
Continuous Random Variable:
A random variable X which assumes any value of real set R or within any interval of real set R, is called continuous random variable.
Probability Density Function:
A function for obtaining probability that a continuous random variable assumes value between specified interval is called probability density function of continuous random variable. It is denoted by f(x).
- The probability that the value of random variable lies within the specified interval is non-negative.
- The total probability that the random variable assumes any value within the specified interval is 1.
- f(x) = P [a < x < b]
- The probability that X assumes a definite value a is zero, i.e., P (x = a) = 0.
∴ P [a < x < b] = P [a ≤ x ≤ b]
Normal Variable:
The continuous random variable X is called normal variable. Its probability density function is denoted by f(x).
Normal Distribution:
The distribution of a normal variable X is called normal distribution. It is expressed by symbol N (µ, σ2); where, µ = mean of normal variable X and σ2 = variance of normal variable X. Normal distribution is symmetrical distribution.
Probability Density Function of Normal Distribution:
The probability density function f(x) of normal distribution is defined as follows:
f(x) = \(\frac{1}{\sigma \cdot \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}\) v a , -∞ < x < ∞
Where, x = Value of random variable X
H = Mean of normal distribution
a = Standard deviation of normal distribution
π = Constant = 3.1416
e = Constant = 2.7183
µ and σ are the parameters of the normal distribution.
Normal Curve:
The curve obtained by plotting the values of probability density function f(x) corresponding the different values of normal random variable X is called normal curve. The normal curve is completely bell-shaped.
Standard Normal Variable and Standard Normal Distribution:
Standard Normal Variable:
If a random variable X is a normal variable with mean fi and standard deviation σ, then the random variable Z = \(\frac{X-\mu}{\sigma}\) is called the standard normal variable. It is free from unit of measurement.
Standard Normal Distribution:
The probability distribution of standard normal variable Z is called standard normal distribution. It’s density function is as follows:
f(z) = \(\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} z^{2}}\), -∞ < z < ∞; where z = \(\frac{x-\mu}{\sigma}\)
Mean of standard normal distribution = 0, Variance = 1 ∴ S.d. = 1
Standard Normal Curve:
The curve of probability density function f(z) of standard normal variable Z is called standard normal curve. It is completely bell-shaped.
Area Under Standard Normal Curve:
- Total area of the region bounded between standard normal and X-axis is = Total Probability 1.
- Standard normal curve is symmetric about Z = 0. Hence, the area of the standard normal curve on both sides of vertical line Z = 0 is equal to 0.5.
- P [0 ≤ Z ≤ a] = Area of standard normal curve bounded by X-axis and vertical lines Z = 0 and Z = a.
Use the ready-made tables of area under the standard normal curve. These tables show the area or probability of standard normal variable Z for the value between
P [μ ≤ X ≤ a] = P [0 ≤ z ≤ z1] = Area under the standard normal curve bounded by X-axls and vertical lines Z = 0 and Z = z1.
( 5 ) Standard normal is completely symmetric about mean (i.e., Z = 0). Hence,
P [-z1 ≤ Z ≤ 0] = P [0 ≤ Z ≤ z1] = 0.5.
( 6) Probability indicates the area under the standard normal curve, while z1 indicates a value of standard normal variable.
Standard score or Z-score:
For the given value x of a normal variable X and for the given values of p and a, the value of Z is called standard score or Z-score.
Symbols:
- X = Normal variate
- x = Value of normal variate X
- Z = Standard normal variate
- z = Value of standard normal variate Z
- μ = Mean (Mu)
- σ = Standard deviation (Sigma)
- f(x) = Probability density function
- π = Constant
- e = Constant
- p = Probability
Formulae:
1. Probability density function of Normal Distribution:
f(x) = \(\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}\) , -∞ < X < ∞
Where, x = Value of normal variate X
μ = Mean of normal distribution
σ = Standard deviation of normal distribution
π = Constant = 3.1416
e = Constant = 2.7183
2. Standard Normal Variate:
Z = \(\frac{\mathrm{x}-\mu}{\sigma}\)
Where, X = Normal variate
μ = Mean of normal distribution
σ = Standard deviation of normal distribution
3. Probability density function of Standard
Normal Distribution:
f(z) = \(\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} z^{2}}\), -∞ < z < ∞
Where, z = \(\frac{x-\mu}{\sigma}\)
4. Parameters of Normal Distribution:
μ = Mean, σ = Standard deviation
5. Mean and S.d. of Standard Normal Distribution:
Mean = 0, S.d. = 1
Hence, variance = 1
6. For Normal Distribution:
- X = M = M0
- Q3 – M = M – Q1
∴ M = \(\frac{\mathrm{Q}_{3}+\mathrm{Q}_{1}}{2}\) - Skewness = 0
- Q1 ≈ μ – 0.675σ, Q3 ≈ μ + 0.675σ
- Quartile deviation ≈ \(\frac{2}{3}\)σ
∴ \(\frac{Q_{3}-Q_{1}}{2} \approx \frac{2}{3}\)σ - Mean deviation ≈ \(\frac{4}{5}\)σ
- Area of region between μ ± σ = 0.6826
Area of region between μ ± 2σ = 0.9545
Area of region between μ ± 3σ = 0.9973
Area of region between μ + 1.96σ = 0.95
Area of region between μ ± 2.575σ =0.99
7. For Standard Normal Distribution:
- X̄ = M = M0 = 0
- Q1 ≈ – 0.675
- Q3 ≈ 0.675
- Area of region between z = ± 1 = 0.6826
Area of region between z = ± 2 = 0.9545
Area of region between z = ± 3 = 0.9973
Area of region between z = ± 1.96 = 0.95
Area of region between z = ± 2.575 = 0.99