Question

The total area under the normal curve equals:

• A. 0.5
• B. 1
• C. 10
• D. 100

Answer 1

Normal Distribution:-

The normal distribution refers to a family of continuous probability distributions described by the normal equation.

The pdf of normal distribution is defined as

where X is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately 3.14159, and e is approximately 2.71828.

The Normal Curve:-

The graph of the normal distribution depends on two factors - the mean and the standard deviation. The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. All normal distributions look like a symmetric, bell-shaped curve, as shown below.

When the standard deviation is small, the curve is tall and narrow; and when the standard deviation is big, the curve is short and wide (see above)

Probability and the Normal Curve:-

The normal distribution is a continuous probability distribution. This has several implications for probability.

• The total area under the normal curve is equal to 1.
• The probability that a normal random variable X equals any particular value is 0.
• The probability that X is greater than a equals the area under the normal curve bounded by a and plus infinity (as indicated by the non-shaded area in the figure below).
• The probability that X is less than a equals the area under the normal curve bounded by a and minus infinity (as indicated by the shaded area in the figure below).

Additionally, every normal curve (regardless of its mean or standard deviation) conforms to the following "rule".

• About 68% of the area under the curve falls within 1 standard deviation of the mean.
• About 95% of the area under the curve falls within 2 standard deviations of the mean.
• About 99.7% of the area under the curve falls within 3 standard deviations of the mean.

Hence, The Answer is

The total area under the normal curve is equal to 1.