Calculus with Algebra and Trigonometry II Lecture 21 Probability applications Apr 16, 215 Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 1 / 1
Histograms The distribution of ages in the US in 199 is given in the following table Age -2 2-4 4-6 6-8 8-1 Percent 3 31 24 14 1 This can be represented by the histogram below. The percent of the population in an interval corresponds to the area. M217L21a.png So the percent of people with ages between 2 and 6 = 55%, the percentage less than 1 = 3/2% and the percent between 75 and 85 = 14/4 + 1/4 = 4.25%. Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 2 / 1
To get a better estimate of the population distribution we make the age ranges smaller -1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 15 15 16 15 13 11 9 5 1 M217L21b.png We can keep on decreasing the size of the interval until the distribution is approximately continuous. Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 3 / 1
Probability density function If we approximate the histograms by a continuous function p(x) it is called a probability density function. The probability of x being between a and b is the area under the curve y = p(x) Probability (a x b) = b a p(x) dx The fraction of the population cannot be negative and the fraction of the population between and is one, thus p(x) must satisfy p(x) p(x) dx = 1 Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 4 / 1
Example We will approximate the US age distribution in 199 by the pdf given below M217L21c.png There are no people over 15 so 15 p(t) dt = 1 4(.15) + 1 2 (.15)a = 1 1 16 (.15)a =.4 a = 2 3 53.3 Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 5 / 1
The slope of the line from (4,.15) to (93.3,) is slope =.15 53.3 2.8 1 4 So the probability density function is {.15 t 4 p(t) = 2.8 1 4 (t 93.3) 4 t 93.3 Note that in this model there are no people over 93.3 years old Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 6 / 1
The Median One way to characterise the average value is the median. The median is the value so that exactly half the population values are less than it. If M is the median it satisfies In our example So the median is M M p(x) dx =.5 p(t) dt =.5.15M =.5 M =.5 33.3 years.15 Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 7 / 1
The Mean Another measure of an average value is the mean, µ, to calculate it you add up all the values in the population and divide by the total number of values thus Mean age = sum of ages of everybody in the US number of people in the US To calculate this using our pdf. Let N be the number of people in the US. The percentage of the population with ages between t and t + deltat is the area under the graph of p(t) and is approximately p(t) t and the number of people with ages between t and t + t is approximately p(t) t. These people have age approximately equal to t, so the sum of the ages of these people is approximately Ntp(t) t Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 8 / 1
As we consider the limit t Sum of ages of all the people in the US = N so the mean is then µ = Sum of ages of all the people in the US N In general for any pdf p(x) the mean is For our example µ = = 15 [.15t 2 2 tp(t) dt = ] 4 4 µ = xp(x) dx t(.15) dt + 2.8 1 4 [ t 2 93.3 2 93.3t 4 ] 93.3 4 = 15 15 tp(t) dt tp(t) dt t( 2.8 1 4 (t 93.3)) dt 36 years Calculus with Algebra and Trigonometry II Lecture 21Probability Apr applications 16, 215 9 / 1
The Standard Deviation The standard deviation is a measure of how spread the data is about the mean. The deviation is the difference between the data valuse and the mean deviation = x µ The standard deviation, σ, is obtained by adding the square of the deviations, dividing by the number of data values and then taking the square root. For a continuous data set with pdf p(x) the standard deviation is defined by For our example σ 2 = 4 σ 2 = (x µ) 2 p(x) dx 93.3.15(t 36) 2 dt 2.8 1 4 (t 36) 2 (t 93.3) dt 484 σ 22 4 Calculus with Algebra and Trigonometry II Lecture 21ProbabilityApr applications 16, 215 1 / 1
Cumulative distribution function Another way of describing the distribution of ages is the cumulative distribution function (cdf), P(t) it is defined to P(t) = Probability (x < t) = t It is the antiderivative of p(x) with P() =. p(x) dx For our example {.15t t 4 P(t) =.6 2.8 1 4 (.5t 2 93.3t) 4 t 93.3 M217L21d.png Calculus with Algebra and Trigonometry II Lecture 21ProbabilityApr applications 16, 215 11 / 1
The normal distribution The most important pdf in applications is the normal (gaussian) distribution. p(x) = 1 σ /2σ2 e (x µ)2 2π µ is the mean and σ is the standard deviation. 1 σ 2π e (x µ)2 /2σ 2 dx = 1 Letting (x µ)/σ = u leads to the equation e u2 /2 du = 2π Calculus with Algebra and Trigonometry II Lecture 21ProbabilityApr applications 16, 215 12 / 1
From the shape of the pdf this distribution is also called the bell curve Empirical_Rule.PNG The distribution is symmetric about the mean with the following rules of thumb 5% is within 2σ/3 of the mean 68% is within σ of the mean 95% is within 2σ of the mean 99.7% is within 3σ of the mean Calculus with Algebra and Trigonometry II Lecture 21ProbabilityApr applications 16, 215 13 / 1