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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 conditions for probability density function for continuous variable are as follows:

• The probability that the value of random variable X lies within the specified interval must be non-negative (positive).
• The total probability that the random variable X assumes any value in the specified interval must be 1.

The normal curve is drawn by considering different values of normal random variable X and its respective values of probability density function f(x).

Here, µ = 10; σ = 6

Estimated value of quartile deviation:
In normal distribution, Quartile deviation ≈ \(\frac{2}{3}\) σ
Putting, σ = 6
Quartile deviation ≈ \(\frac{2}{3}\) ≈ = 4
Hence, the estimated value of quartile deviation obtained is 4.

If X is normal variable with mean = µ and standard deviation = σ, then the random variable Z = \(\frac{X−μ}σ\) is called the standard normal variable. It is free from units of measurement.

The probability density function of Z is as follows :

f(z) = \(\frac{1}{\sqrt{2π}}e^{-\frac{1}{2}z^2}\), – ∞ < z < ∞

Here, mean deviation = 8

In normal distribution, mean deviation ≈ \(\frac{4}{5}\) σ

∴ 8 ≈ \(\frac{4}{5}\) σ

∴ σ ≈ \(\frac{8×5}{4}\) ≈ 10

Hence, the standard deviation obtained is 10.

The shape of normal curve is completely bell-shaped.

The shape of standard normal curve is complete bell-shaped. It is symmetric about the value of variable Z = 0.

If a random variable X is a normal variable with mean fi and standard deviation σ, then the random variable Z =\(\frac{ X−μ}{σ}\)σ is called the standard normal variable. It is free from unit of measurement.

The skewness of normal distribution is zero.

For Z = 0, the standard normal curve is symmetric on both the sides.

Here, µ = 45; σ = 10, x – 60

∴ Z = \(\frac{x−μ}{σ} = \frac{60−45}{10}\) = 1.5

Hence, Z-score = 1.5

The curve of probability density function f(z) of standard normal variable Z is called standard normal curve. It is completely bell-shaped.

The probability distribution of standard normal variable Z is called standard normal distribution. It’s density function is as follows:

\(f(z) = \frac{1}{2π}e^{−\frac{1}{2}z^2}\,-∞ < z < ∞\)

where z = \(\frac{x−μ}{σ}\)

Mean of standard normal distribution = 0, Variance = 1 ∴ S.d. = 1

• 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.

“Standard score is independent of unit of measurement.” This statement is true.

Mean = 0, S.d. = 1
Hence, variance = 1

\(Z=\frac{x−μ}σ\)

Where, X = Normal variate
μ = Mean of normal distribution
σ = Standard deviation of normal distribution

μ = Mean, σ = Standard deviation

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.

\(f(x) = \frac{1}{\sqrt{σ2π}}e^{−\frac{1}2(\frac{x−μ}σ)^2} , -∞ < X < ∞\)

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