Advanced High-School Mathematics

SECTION 6.3 Parameters and Statistics 365

9. LetXbe an exponential random variable with failure rateλ, and
   letY =X^1 /α, α >0. Using the idea developed on page 353, com-
   pute the density function forY. This gives the so-calledWeibull
   distribution.

6.3 Parameters and Statistics

Suppose that we have a continuous random variableXhaving density
functionfX. Associated with this random variable are a fewparame-
ters, themean(and also themedian and themode) and the vari-
anceofX. In analogy with discrete random variables they are defined
as follows.

Mean of X. We set

E(X) = μX =

∫∞

−∞

xfX(x)dx.

Median of X. This is just the half-way point of the distribution,

that is, ifmis the median, we haveP(X≤m) =^12 =P(X≥m).

In terms of the density function, this is just the valuemfor which

∫m

−∞fX(x)dx =

1

2

.

Mode ofX. This is just the value ofxat which the density function

assumes its maximum. (Note, then, that the mode might not be

unique: a distribution might be “bimodal” or even “multimodal.”)

Themean,median, andmodemeasure “central tendency.”

Variance of X. We set

Var(X) = σ^2 X = E((X−μX)^2 ).

As we shall see below,