Fundamentals of Probability and Statistics for Engineers

78 Fundamentals of Probability and Statistics for EngineersAnswer: the average value required in this example is a conditional mean of Rgiven the event A. Although no formal definition is given, it should be clear thatthe desired average is obtained fromEfRjAg ˆZ 52This integral can be evaluated when f R (r) is specified.48rf R …rjA†dr ˆZ 52Two other quantities in common usage that also give a measure of centralityof a random variable are its median and mode.A median of X is any point that divides the mass of the distribution into twoequal parts; that is, x 0 is a median of X ifP…X x 0 †ˆ12 :The mean of X may not exist, but there exists at least one median.In comparison with the mean, the median is sometimes preferred as ameasure of central tendency when a distribution is skewed, particularly wherethere are a small number of extreme values in the distribution. For example, wespeak of median income as a good central measure of personal income for apopulation. This is a better average because the median is not as sensitive toa small number of extremely high incomes or extremely low incomes as isthe mean.where is a positive constant. The random variable T is called the lifetime ofthe atom, and a common average measure of this lifetime is called the half-life,which is defined as the median of T. Thus, the half-life, is found from48rf R …r†dr:cEx ample 4. 4. Let T be the time between emissions of particles by a radioactiveatom. It is well established that T is a random variable and that it obeysan exponential distribution; that is,f T …t† ˆ e t ; for t 0;0; elsewhere;Z 0f T …t†dt ˆ 12 ;or ˆ ln2 :TLFeBOOK