− () x x = e, x ≥ 0 to a data set can sometimes increase with the square of N, the

f X − () x x = e, x ≥ 0 to a data set can sometimes increase with the square of N, the

  Determine the probability distribution for the following:

  number of rows of data. Suppose that for a particular algo-

  (a) Y = X 2 2 (b) Y = X 12 (c) Y = ln X rithm, the computation time is approximately T =. 0 004 N sec-

  5-84.

  The velocity of a particle in a gas is a random vari-

  onds. Although the number of rows is a discrete measurement,

  able V with probability distribution

  assume that the distribution of N over a number of data sets can

  fv V av e 2 − () bv = v> 0 be approximated with an exponential distribution with a mean

  where b is a constant that depends on the temperature of the gas

  of 10,000 rows. Determine the probability density function and

  and the mass of the particle.

  the mean of T.

  (a) Determine the value of the constant a.

  5-90. Power meters enable cyclists to obtain power meas-

  (b) The kinetic energy of the particle is W mV 2 =

  2. Determine

  urements nearly continuously. The meters also calculate the

  the probability distribution of W.

  average power generated over a time interval. Professional

  Section 5-6Moment-Generating Functions

  riders can generate 6.6 watts per kilogram of body weight for

  power is computed as the fourth root of the mean of Y = 4 X .

  extended periods of time. Some meters calculate a normal-

  Determine the following:

  ized power measurement to adjust for the physiological effort

  (a)

  Mean and standard deviation of X

  required when the power output changes frequently. Let the

  (b)

  fy Y ()

  random variable

  X denote the power output at a measure-

  (c)

  Mean and variance of Y

  ment time and assume that

  X has a lognormal distribution

  (d)

  Fourth root of the mean of Y

  with parameters

  θ=.

  5 2933 and 2

  ω =. 0 00995 . The normalized

  (e) Compare [( EX 4 )] 14 to E X ( ) and comment.

  5-6 Moment-Generating Functions

  Suppose that X is a random variable with mean μ. Throughout this book we have used the idea

  of the expected value of the random variable X, and in fact EX () = μ. Now suppose that we are interested in the expected value of a function of X, r gX () = X . The expected value of this

  function, or r EgX [ ( )] = EX ( ) , is called the rth moment about the origin of the random vari-

  able X, which we will denote by μ r ′ .

  Definition of Moments

  about the Origin

  The rth moment about the origin of the random variable X is

  ⎧ ⎪ ∑ xfx ( ),

  X discrete

  X continuous

  ⎩ −∞

  Notice that the first moment about the origin is just the mean, that is, EX () = μ ′ 1 . Fur- thermore, since the second moment about the origin is EX () 2 = μ′ , we can write the variance

  of a random variable in terms of origin moments as follows:

  σ 2 = EX ( ) [ ( )]

  − ′ = μ −

  The moments of a random variable can often be determined directly from the definition in Equation 5-32, but there is an alternative procedure that is frequently useful that makes use of

  a special function.

  Definition of a Moment-Generating

  The moment-generating function of the random variable X is the expected value of

  Function

  e tX and is denoted by Mt

  X ( ). That is,

  ⎧

  ⎪ ∑ efx ( ),

  X discrete

  X continuous

  ⎩ −∞

  The moment-generating function Mt X ( ) will exist only if the sum or integral in the above defi-

  nition converges. If the moment-generating function of a random variable does exist, it can be used to obtain all the origin moments of the random variable.

  Let X be a random variable with moment-generating function Mt X ( ). Then

  dMt r

  μ′r =

  X ()

  t

  0 (5-34)

  dt

  r

  Chapter 5Joint Probability Distributions

  Assuming that we can differentiate inside the summation and integral signs,

  X ⎪ discretee ∑

  X continuous

  ⎩ −∞ Now if we set t = 0 in this expression, we fi nd that