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CC9 - Probability
CONTINUOUS PROBABILITY DISTRIBUTION
In the previous section, we investigated probability distributions of discrete random
variables, that is, random variables whose sample space contains a countable number of
outcomes. In the discrete case, the number of outcomes in the sample space can be either
finite or countably infinite.
In this section, as the title suggests, we are going to investigate probability distributions of
continuous random variables, that is, random variables whose sample space contains an
infinite interval of possible outcomes.
Make sure your calculus skills of integration and differentiation are up to snuff.
Definition: Continuous Random Variable
- A random variable X is said to be continuous if it can assume any value or infinite
number of possible values in a given interval. This contrasts with the definition of a
discrete random variable which can only assume discrete values.
a. Rainfall
b. Temperature
c. Weight in pounds
- This kind of random variable is called a continuous random variable and it is
characterised or can be modelled, not by probabilities mass function ๐(๐ = ๐ฅ) (as
was the case with a discrete random variable), but by a function called the
probability density function (pdf for short).
- In experiments of this kind, we never determine the probability that the random
variable assume a particular value, but only calculate the probability that it lies
within a given range of values say (๐, ๐), that is finding ๐(๐ < ๐ < ๐) , where a and
b are some constants.
Probability Density Function
- The probability density function ("p.d.f.") of a continuous random variable X with
support ๐ is an integrable function ๐(๐ฅ) satisfying the following:
- ๐(๐ฅ) > 0 , for all ๐ฅ โ ๐
โ
โโ
Example: Identify the following function whether it is a valid probability density function or
not.
๐ฅ
2
2
10
3
2
๐ฅ
3
4
Note:
โ ๐(๐ฅ) represents the height of the curve at point x, thus ๐
โ In continuous rv, it is the areas under the curve that define the probabilities.
โ The probability the random variable X is exactly equal to any specific value is 0 that
is ๐(๐ฅ = ๐) = 0. Thus ๐
Cumulative Distribution Function
- CDF accumulates all the probabilities less than or equal to x, which can be written
as ๐น(๐ฅ) = ๐(๐ โค ๐ฅ).
- CDF show the probability of all x-values up to a certain x occurring.
- The cumulative distribution function of a continuous random variable X is defines as
๐ฅ
โ โ
โข Where t is called the dummy variable because x is one of our limits of integration.
Properties of CDF
- It must be non-decreasing (monotonic)
- lim
๐ฅโโโ
๐ฅโ+โ
CONTINUOUS UNIFORM PROBABILITY DISTRIBUTION
- The Uniform or Rectangular distribution, where the random variable X is restricted
to a finite interval [a, b] and f(x) is constant over the range of possible values of x
often defined by a function of the form:
where: a = smallest value the variable can assume
b = largest value the variable can assume
- The expected value and variance of the continuous uniform probability distribution:
2
Example: Slater customers are charged for the amount of salad they take. Sampling
suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15
ounces.
a. Determine the probability density function.
b. Draw a graph of f(x).
c. What is the probability that a customer will take between 12 and 15 ounces of
salad?
d. Find the expected value and variance of X.
Example: The amount of time a person must wait for a train to arrive in a certain town is
uniformly distributed between 0 and 40 minutes.
a. Determine the probability density function.
b. Draw a graph of f(x).
c. What is the probability that a person must wait less than 8 minutes?
d. What is the probability that a person must wait more than 30 minutes?
e. Find the expected value and variance of X.
The Normal Distribution
The normal distribution is the most widely used model for the distribution of a random
variable. There is a very good reason for this. Practical experiments involve measurements
and measurements involve errors. Errors are unavoidable and are usually the sum of
several factors. It has been used in a wide variety of applications including: heights of
people, rainfall amounts, test scores, etc.
The probability density function of a normal distribution with mean ฮผ and variance ๐
2
is
given by the formula
โ
( ๐ฅโ๐
)
2
/ 2 ๐
2
Characteristics of a normal distribution:
- The distribution is symmetric; its skewness measure is zero.
- The entire family of normal probability distributions is defined by its mean and its
standard deviation
- The highest point on the normal curve is at the mean, which is also the median and
mode.
- The standard deviation determines the width of the curve: larger values result in
wider, flatter curves.
- It is asymptotic along x-axis.
- Probabilities for the normal random variable are given by areas under the curve.
The total area under the curve is 1 (.5 to the left of the mean and .5 to the right).
A normal distribution has mean ฮผ and variance ๐
2
. A random variable X following this
distribution is usually denoted by N(ฮผ, ๐
2
) and we often write ๐ ~ ๐ (๐ , ๐
2
Clearly, since ฮผ and ๐
2
can both vary, there are infinitely many normal distributions and it
is impossible to give tabulated information concerning them all.
Without tabulated data concerning the appropriate normal distribution we cannot easily
answer this question (because the integral used to calculate areas under the normal curve
is intractable.) Since tabulated data allow us to apply the distribution to a wide variety of
statistical situations, and we cannot tabulate all normal distributions, we tabulate only one
- the standard normal distribution - and convert all problems involving the normal
distribution into problems involving the standard normal distribution.
Standard Normal Distribution
- The standard normal distribution has a mean of zero and a variance of one.
- The probability density function of standard normal distribution is given by:
โ
( ๐ฅ
)
2
/ 2
The Exponential Distribution
The exponential probability distribution is useful in describing the time it takes to
complete a task. The exponential distribution is a continuous distribution that is commonly
used to measure the expected time for an event to occur.
The time X we need to wait before an event occurs has an exponential distribution if the
probability that the event occurs during a certain time interval is proportional to the length
of that time interval.
The probability density function (pdf) of an exponential distribution is
โ๐๐ฅ
The parameter ๐ is called the rate parameter. It is the inverse of the expected duration
(๐). Where ๐ is the average/mean.
The cumulative distribution function of an exponential distribution can be written as:
โ๐๐ฅ
The expected value and variance of an exponential random variable X is given by:
2
2
Example:
- The time between arrivals of cars at Alโs full- service gas pump follows an
exponential probability distribution with a mean time between arrivals of 3 minutes.
Al would like to know the probability that the time between two successive arrivals
will be 2 minutes or less.
- Assume that the length of a phone call in minutes is an exponential random variable
X with parameter ๐ =
1
10
. If someone arrives at a phone booth just before you arrive,
find the probability that you will have to wait
a. Less than 5 minutes
b. Greater than 10 minutes
c. Between 5 and 10 minutes
