Probability Distribution Curve

The Normal distribution or Gaussian distribution is the continuous probability distribution that often gives a good description of data that cluster around the mean. The graph of the associated probability density function is a bell-shaped with a peak at the mean and is known as the Gaussian function or bell curve. The shape of normal distribution resembles that of a bell. So it is referred as the "bell curve".

Definition of probability distribution curve:

A continuous random variable X is said to follow a probability distribution curve with parameter μ and σ (or μ and σ2) if the probability function is
Y = [1/σ *   ] * e-(x - μ)2/2σ2
X -μ(μ, σ) denotes that the random variable X follows bell curve distribution  Even we can write the probability distribution curve as X-μ(μ, σ2) symbolically. In this case the parameters are mean and standard deviation with mean μ and standard deviation σ. Where X = normal random variable, μ= mean, σ = standard deviation, π = 3.14159, e = 2.71828.
A probability distribution is evaluated by the probability of random variable X in open interval (-∞, x), which is given by,
F(x) = P [X ≤ x]
“Probability distribution curve is often called as normal distribution or bell curve or Gaussian curve”
Types of probability distribution:

Discrete probability distribution,
Continuous probability distribution.

Discrete probability distribution:

If the cumulative distribution function of a probability distribution increases in jumps then the probability distribution is said to be discrete.
The discrete probability distribution with the probability mass function p is given by,
P [X = x] = p(x).

Continuous probability distribution:
If the cumulative distribution function F(x) = µ (-∞, x) is continuous then the probability distribution function is called as continuous probability distribution function. These are characterized by the probability density function f given by,

x
F(x) = µ (-∞, x) = `int` f (t) dt
-∞

Example problem based on probability distribution curve:

The mean score of 1000 students for an examination is 34 and S.D is 16.  How many candidates can be expected to obtain marks between 30 and 60 assuming the normality of the distribution?

Solution: The given data are μ = 34, σ = 16, N = 1000

We have to find the probability of marks between P (30 < X < 60)

We can use the formula, Z = `(X - mu)/sigma`

X = 30,

Z1 = `(30 - mu)/sigma`

=>`(30 - 34)/16`

Z1 = `(-4)/(16)`

Z1 = – 0.25

Z2 = `(60-34)/16`

= `(26)/(16)`

= 1.625


So the value of z2 = 1.63 (approximate)

P(−0.25 < Z < 1.63) =P(0 < Z< 0.25) + P(0 < Z < 1.63) (due to symmetry)

= 0.0987 + 0.4484 = 0.547

No of students scoring between 30 and 60 is, 0.5471 × 1000 => 547.