Expected Value and Variance Expected Value We have seen that for a discrete random variable, that the expected value is the sum of all xP(x). For continuous random variables, P(x) is the probability density function, and integration takes the place of addition.
Remark: Integration by parts and/or substitution will typically be used to perform the integration.
Example Find the expected value of the density function defined by f(x) = sin(x) 0 < x < p/2
Solution We compute the integral
We use integration by parts with
u = x dv =
sin(x) dx We have
Variance and Standard Deviation The variance formula for a continuous random variable also follows from the variance formula for a discrete random variable. Once again we interpret the sum as an integral.
Example Use a calculator to find the variance and standard deviation of the density function f(x) = 6x  6x^{2} 0 < x < 1
Solution We first need to find the expected value. We have
Now we can compute the variance
Finally the standard deviation is the square root of the variance or s = 0.22 The Median The expected value is what you are used to as the average. Another useful number is the median which gives the halfway point. Since the total area under a probability density function is always equal to one, the halfway point of the data will be the xvalue such that the area from the left to the median under f(x) is equal to 1/2.
Example Find the median of the probability density function
1
Solution We integrate
Now we set the above equal to 1/2 and solve.
m^{3}
 27 = 94.5 m = 4.95 The Uniform Density Function If every interval of a fixed length is equally likely to occur then we call the probability density function the uniform density function. It has formula
1
Example The distribution of insects along a fallen log of length twenty feet is uniform. Find the standard deviation for this distribution.
Solution First notice that the density function is given by
1 The expected value is given by
Next, we find the variance. We have
The standard deviation is the square root s = 5.77 The Normal Distribution The most important distribution for working with statistics is called the normal distribution. If with mean m and standard deviation s the density function is given by
Exercise Show that the expected value of the normal distribution is m. The Exponential Distribution Another important distribution is call the exponential distribution. It has density function
Exercise Find the mean and standard deviation of the exponential distribution.
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