- •4.1. Introduction
- •4.2. Areas under continuous probability density functions
- •4.3. The normal distribution
- •1. The total area under the curve is 1.0.
- •4.3.1. Cumulative distribution function of the
- •4.4. The standard normal distribution
- •4.5. Standardizing a normal distribution
- •4.6. The normal distribution approximation to the binomial
- •4.7. The exponential probability distribution
Chapter 4
Continuous random variables and their probability distributions
4.1. Introduction
U p to this point, we have limited our discussion to probability distributions of discrete random variables. Recall that a discrete random variable takes on only some isolated values, usually integers representing a count. We now turn our attention to the probability distribution of a continuous random variable- one that can ideally assume any value in an interval. Variables measured on an underlying continuous scale, such as weight, strength, life length, and temperature, have this feature.
Figure 4.1 displays the histogram and
polygon for some continuous data set.
The smoothed polygon is an approximation
of the probability distribution curve of the continuous random variable X. The probability distribution curve of a continuous random variable is also called
its probability density function.
The probability density function, denoted by possesses the following characteristics:
1. for all x.
2 . The area under the probability density function over all possible values of the random variable X
is equal to 1.
3. Let a and b be two possible values of the random variable X, with . Then the probability that X lies between a and b is the area under the density function between a and b.(Fig.4.2)
4. The cumulative distribution function is the area under the probability density function up to
where is the minimum value of the random variable X.
4.2. Areas under continuous probability density functions
Let X be a continuous random variable with probability density function and cumulative distribution function . Then:
1. The total area under the curve is 1.
2. The area under the curve to the left of is ,
where is any value that the random variable X can take.
T he area under the probability distribution curve of a continuous random variable between any two points is between 0 and 1, as shown in Figure 4.3.
The total area under the probability distribution curve of a continuous random variable is always 1.0 or 100% as shown in Figure 4.4.
Remark:
The probability that a continuous random variable x assumes a single value is always zero.
This is because the area of a line, which represents a single point,
is zero. (Fig.4.5)
In general, if a and b are two of the values that X can assume, then,
and .
When determining the probability of an interval a to b, we need not be concerned if either or both end points are included in the interval. Since the probabilities of and are both equal to 0,
.
Exercises
1. Which of the functions sketched in a-d could be a probability density function for a continuous random variable? Why or why not?
f
(x)
1
2
x
0
b)
2. Determine the following probabilities from the curve diagrammed in exercise 1(a).
a) b)
c) d)
3. For the curve graphed in exercise 1(c) which of the two intervals or is assigned a higher probability?
4. The time it takes for a TV repair master to finish his job (in hours) has a
density function of the form
a) Determine the constant c.
b) What is the probability that a TV repair master will finish the job in less than 75 minutes? Between and 2 hours?
5. Suppose that the loss in a certain investment, in thousands of dollars, is a continuous random variable X that has a density function of the form
a) Calculate the value of k.
b) Find the probability that the loss is at most $500.
6. Let the random variable X has probability density function
a) Draw the probability density function
b) Show that the density function has the properties of a proper probability density function
c) Find the probability that X takes a value between 0.5 and 1.5.
Answers
2. a) 0.25; b) 0.25; c) 0.25; d) 0; 3. The interval 1.5 to 2 has higher probability; 4. a) -6; b) 5/32 ; 1/2; 5. a) -1/2; b) 3/16; 6. c) 0.75.