# Assignment on Normal Distribution and Continuous Random Variables

**Topics:**Normal distribution, Standard deviation, Probability density function

**Pages:**5 (542 words)

**Published:**October 10, 2014

CONTINUOUS RANDOM VARIABLES

A random variable whose values are not countable is called a _CONTINUOUS RANDOM VARIABLE._

THE NORMAL DISTRIBUTION

The _NORMAL PROBABILITY DISTRIBUTION_ is given by a bell-shaped(symmetric) curve.

THE STANDARD NORMAL DISTRIBUTION

The normal distribution with

and

is called the _STANDARD NORMAL DISTRIBUTION._

Example 1: Find the area under the standard normal curve

between z = 0 and z = 1.95

from z = -2.17 to z = 0

Area to the right of z = 2.32

Area to the left of z = -1.54

Example 2a: Find the following probabilities for the standard normal curve.

(a)

(b)

(c)

Example 2b: Find the value of _k_ if

(a)

(b)

(c)

STANDARDIZING A NORMAL DISTRIBUTION

Example 3: Let x be a normal random variable with its mean equal to 40 and standard deviation equal to 5. Find the following probabilities for this normal distribution.

(a)

(b)

(c)

(d)

Example 4: Lengths of metal strips produced by a machine are normally distributed with mean length of 150cm and a standard deviation of 10cm.

Find the probability that the length of a randomly selected strip is

(a) shorter than 165cm,

(b) within 5cm of the mean.

Example 5: The time taken by the milkman to deliver to the High Street is normally distributed with a mean of 12 minutes and a standard deviation of 2 minutes. He delivers milk every day. Estimate the number of days during the year when he takes

(a) longer than 17 minutes,

(b) less than ten minutes,

(c) between 9 and 13 minutes.

Example 6: The height of female students at a particular college are normally distributed with a mean of 169cm and a standard deviation of 9 cm.

(a) Given that 80% of these female students have a height less than _h_ cm, find the value of _h._

(b) Given that 60% of these female students have a height greater than _s_ cm, find the value of _s_.

Example 7: The marks of 500 candidates in an examination are normally distributed with a mean of 45 marks and a standard deviation of 20 marks.

(a) Given that the pass mark is 41, estimate the number of candidates who passed the examination.

(b) If 5% of the candidates obtain a distinction by scoring _x_ marks or more, estimate the value of _x_.

Example 8: The lengths of certain item follow a normal distribution with mean

cm and standard deviation 6 cm. It is known that 4.78% of the items have length greater than 82cm. Find the value of the mean

.

Example 9: The masses of boxes of oranges are normally distributed such that 30% of them are greater than 4.00kg and 20% are greater than 4.53 kg. Estimate the mean and standard deviation of the masses.

Normal probability distribution

1. The total area under the curve is 1 or 100%

2. The curve is symmetric about the mean.

3. Two tails of the curve extend indefinitely.

Standard deviation = � EMBED Equation.3 ���

Mean = � EMBED Equation.3 ���

z Values or z scores

The units marked on the horizontal axis of the standard normal curve are denoted by z and called the z values or z scores.

Converting an x value to a z value

For a normal random variable x, a particular value of x can be converted to its corresponding z value:

� EMBED Equation.3 ���

� PAGE �1�

_1288036674.UNKNOWN

_1288417991.UNKNOWN

_1462022587.UNKNOWN

_1462022985.UNKNOWN

_1462022803.UNKNOWN

_1288418041.UNKNOWN

_1288420124.UNKNOWN

_1288417501.UNKNOWN

_1288417541.UNKNOWN

_1288036704.UNKNOWN

_1288416916.UNKNOWN

_1288006205.UNKNOWN

_1288036634.UNKNOWN

_1288005109.UNKNOWN

_1288006188.UNKNOWN

_1288005055.UNKNOWN

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