(c) No accidents (d) Fewer than five accidents lity Distributions 4.4.4 In a study of the effectiveness of an insecticide against a certain insect, a large area of land was sprayed. Later the area was examined for live insects by randomly selecting squares and counting the number of live insects per square. Past experience has shown the average number of live insects per square after spraying to be .5. If the number of live insects per square follows a Poisson distribution, find the probability that a selected 10 square will contain: (a) Exactly one live insect (b) No live insects (c) Exactly four live insects (d) One or more live insects 46 The No 4.5 Continuous Probability Distributions The probability distributions considered thus far, the binomial and the Poisson, are distributions of discrete variables. Let us now consider distributions of continuous random variables. In Chapter 1, we stated that a continuous variable is one that can assume any value within a specified interval of values assumed by the variable. Consequently, between any two values assumed by a continuous variable, there exist an infinite number of values. To help us understand the nature of the distribution of a continuous random variable, let us consider the data presented in Table 1.4.1 and Figure 2.6.1. In the table, we have 189 values of the random variable, age. The histogram of Figure 2.6.1 was constructed by locating speceilicd Poms on a line representing the measurement of interest and erecting a series of rectangies. whose widths were the distances between two specified points on the line, and whose heights represented the number of values of the variable falling between the two specified points. The intervals defined by any two consecutive specified points we called class intervals. As was noted of the variable between the horizontal scale boundaries of these subareas. This provides a way be calculated: merely determine the proportion of the histogram's total area falling between the whereby the relative frequency of occurrence of values between any two specified points can specified points. This can be done more conveniently by consulting the relative frequency or cumulative relative frequency columns of Table 2.3.2. and the width of our class intervals is made very small. The resulting histogram could look like Imagine now the situation where the number of values of our random variable is very large that shown in Figure 4.5.1. frequency polygon, clearly we would have a much smoother figure than the frequency polygon If we were to connect the midpoints of the cells of the histogram in Figure 4.5.1 to form a In general, as the number of observations, n, approaches infinity, and the width of the class Figure 4.5.2. Such smooth curves are used to represent graphically the distributions of continuous intervals approaches zero, the frequency polygon approaches a smooth curve such as is shown in Consequences when we deal with probability distri- as was true with the histogram, and ints on the r-axis is equal to cd at the two points m variable is area