Normal Distribution
Normal Distribution
A continuous random variable X having the bell-shaped distribution of Figure is called a normal random
variable. The mathematical equation for the probability distribution of the normal variable depends on the
two parameters μ and σ, its mean and standard deviation, respectively. Hence, we denote the values
of the density of X by n(x; μ, σ).
Probability density function of normal
distribution
Properties of Normal distribution
1. The mode, which is the point on the horizontal axis where the curve is a
maximum, occurs at x = μ.
2. The curve is symmetric about a vertical axis through the mean μ.
3. The curve has its points of inflection at x = μ ± σ; it is concave downward if
μ - σ < X < μ + σ and is concave upward otherwise.
4. The normal curve approaches the horizontal axis asymptotically as we
proceed in either direction away from the mean.
5. The total area under the curve and above the horizontal axis is equal to 1.
6. The mean and variance of n(x; μ, σ) are μ and 𝜎 2 , respectively. Hence, the
standard deviation is σ.
Standard Normal variate
The distribution of a normal random variable with mean 0 and variance 1 is
called a standard normal distribution.
6.8 Given a normal distribution with μ = 30 and
σ = 6, find
(a) the normal curve area to the right of x = 17;
(b) the normal curve area to the left of x = 22;
(c) the normal curve area between x = 32 and x = 41;
(d) the value of x that has 80% of the normal curve
area to the left;
(e) the two values of x that contain the middle 75% of
the normal curve area
6.6 Find the value of z if the area under a standard
normal curve
(a) to the right of z is 0.3622;
(b) to the left of z is 0.1131;
(c) between 0 and z, with z > 0, is 0.4838;
(d) between -z and z, with z > 0, is 0.9500
PRACTICE QUESTIONS
6.7 Given a standard normal distribution, find the
value of k such that
(a) P (Z > k) = 0.2946;
(b) P (Z < k) = 0.0427;
(c) P (-0.93 < Z < k) = 0.7235.
PRACTICE QUESTIONS
6.9 Given the normally distributed variable X with
mean 18 and standard deviation 2.5, find
(a) P (X < 15);
(b) the value of k such that P (X < k) = 0.2236;
(c) the value of k such that P (X > k) = 0.1814;
(d) P (17 < X < 21)
PRACTICE QUESTIONS
6.13 A research scientist reports that mice will live an
average of 40 months when their diets are sharply restricted and then enriched with
vitamins and proteins.
Assuming that the lifetimes of such mice are normally
distributed with a standard deviation of 6.3 months,
find the probability that a given mouse will live
(a) more than 32 months;
(b) less than 28 months;
(c) between 37 and 49 months
PRACTICE QUESTIONS
6.18:The heights of 1000 students are normally distributed with a mean of 174.5
centimeters and a standard deviation of 6.9 centimeters. Assuming that the
heights are recorded to the nearest half-centimeter,
how many of these students would you expect to have
heights
(a) less than 160.0 centimeters?
(b) between 171.5 and 182.0 centimeters inclusive?
(c) equal to 175.0 centimeters?
(d) greater than or equal to 188.0 centimeters?
PRACTICE QUESTIONS
6.21 The tensile strength of a certain metal component is normally distributed with a
mean of 10,000 kilograms per square centimeter and a standard deviation of 100
kilograms per square centimeter. Measurements are recorded to the nearest 50
kilograms per square centimeter.
(a) What proportion of these components exceed 10,150 kilograms per square
centimeter in tensile strength?
(b) If specifications require that all components have tensile strength between 9800
and 10,200 kilograms per square centimeter inclusive, what proportion of pieces
would we expect to scrap?
PRACTICE QUESTIONS
6.22 If a set of observations is normally distributed,
what percent of these differ from the mean by
(a) more than 1.3σ?
(b) less than 0.52σ?
