We can see from the first line of the table that the area to the left of −5.22 must be so close to 0 that to four decimal places it rounds to 0.0000. Similarly, here we can read directly from the table that the area under the density curve and to the left of 2.15 is 0.9842, but −5.22 is too far to the left on the number line to be in the table.
We can see from the last row of numbers in the table that the area to the left of 4.16 must be so close to 1 that to four decimal places it rounds to 1.0000. We obtain the value 0.8708 for the area of the region under the density curve to left of 1.13 without any problem, but when we go to look up the number 4.16 in the table, it is not there. We attempt to compute the probability exactly as in Note 5.20 "Example 6" by looking up the numbers 1.13 and 4.16 in the table. Its area is the base of the rectangle times its height, 10 This is the shaded region in Figure 5.4 "Probability of Waiting At Most 10 Minutes for a Bus". By definition, this probability is the area of the rectangular region bounded above by the horizontal line f ( x ) = 1 ∕ 30, bounded below by the x-axis, bounded on the left by the vertical line at 0 (the y-axis), and bounded on the right by the vertical line at 10. The probability sought is P ( 0 ≤ X ≤ 10 ). See Figure 5.4 "Probability of Waiting At Most 10 Minutes for a Bus". Since the total area under the curve must be 1, the height of the horizontal line is 1/30. The graph of the density function is a horizontal line above the interval from 0 to 30 and is the x-axis everywhere else. Find the probability that a bus will come within the next 10 minutes. Buses run every 30 minutes without fail, hence the next bus will come any time during the next 30 minutes with evenly distributed probability (a uniform distribution). If we think of X as a measurement to infinite precision arising from the selection of any one member of the population at random, then P ( a < X < b ) is simply the proportion of the population with measurements between a and b, the curve in the relative frequency histogram is the density function for X, and we arrive at the definition just above.Įvery density function f ( x ) must satisfy the following two conditions:Ī man arrives at a bus stop at a random time (that is, with no regard for the scheduled service) to catch the next bus. Moreover the total area under the curve is 1, and the proportion of the population with measurements between two numbers a and b is the area under the curve and between a and b, as shown in Figure 2.6 "A Very Fine Relative Frequency Histogram" in Chapter 2 "Descriptive Statistics". There we saw that if we have in view a population (or a very large sample) and make measurements with greater and greater precision, then as the bars in the relative frequency histogram become exceedingly fine their vertical sides merge and disappear, and what is left is just the curve formed by their tops, as shown in Figure 2.5 "Sample Size and Relative Frequency Histograms" in Chapter 2 "Descriptive Statistics". This definition can be understood as a natural outgrowth of the discussion in Section 2.1.3 "Relative Frequency Histograms" in Chapter 2 "Descriptive Statistics". More meaningful questions are those of the form: What is the probability that the commuter's waiting time is less than 10 minutes, or is between 5 and 10 minutes? In other words, with continuous random variables one is concerned not with the event that the variable assumes a single particular value, but with the event that the random variable assumes a value in a particular interval.įigure 5.1 Probability Given as Area of a Region under a Curve If anything the probability should be zero, since if we could meaningfully measure the waiting time to the nearest millionth of a minute it is practically inconceivable that we would ever get exactly 7.211916 minutes. But although the number 7.211916 is a possible value of X, there is little or no meaning to the concept of the probability that the commuter will wait precisely 7.211916 minutes for the next bus. If buses run every 30 minutes without fail, then the set of possible values of X is the interval denoted, the set of all decimal numbers between 0 and 30. For example, suppose X denotes the length of time a commuter just arriving at a bus stop has to wait for the next bus.
This is not the case for a continuous random variable.
#CAN YOU CALCULATE WEIGHTED STANDARD DEVIATION CONTINUOUS TRIAL#
The Probability Distribution of a Continuous Random Variableįor a discrete random variable X the probability that X assumes one of its possible values on a single trial of the experiment makes good sense.
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