STA1101 Normal Distribution and Continuous random variables

# 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

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)

Example 2b: Find the value of *k* if

(a)

# 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...