Page 220 - Applied statistics and probability for engineers

198 Chapter 5/Joint Probability Distributions

Mind-Expanding Exercises

5-126. Show that if X , X ,…, X are independent, selected, without replacement, from the population. Let X ,
1 2 p 1
continuous random variables, P(X ∈ A , X ∈ A ,…, X ,…, X denote the number of items of each type in the sam-
1 1 2 2 2 k
X ∈ A ) = P(X ∈ A )P(X ∈ A ) … P(X ∈ A ) for ple so that X + X , + … + … + X = n. Show that for feasible
p p 1 1 2 2 p p 1 2 k
any regions A , A ,…, A in the range of X , X ,…, X values of n, x , x , …, x , N , N , …, N , the probability is
1 2 p 1 2 p 1 2 k 1 2 k
respectively. ⎞ ⎞
⎛ N 1 ⎞ ⎛ N 2 … ⎛ N k
5-127. Show that if X , X ,…, X are independent random ⎜ ⎟ ⎜ ⎟ ⎠ ⎜ ⎟ ⎠
p
1
2
variables and Y = c X + c X + ... + c X , V Y ( ) = c V X ( ) + P (X = x , X = x ,..., X = x ) = ⎝ x 1 ⎠ ⎝ x 2 ⎝ x n
2
⎛ ⎞
1
1
k
p
p
k
1
1
2
2
1
2
2
1
2
2
c V X 2 ( ) + ... + c V X p ( ) N
p
2
⎜ ⎟
⎝ ⎠
You may assume that the random variables are continuous. n
5-128. Suppose that the joint probability function of 5-131. Use the properties of moment generating functions
the continuous random variables X and Y is constant on to show that a sum of p independent normal random vari-
the rectangle 0 < x < a, 0 < y < b. Show that X and Y are ables with means μ and variances σ for i = 1,2, ...., p has a
2
i
i
independent. normal distribution.
tX
5-129. Suppose that the range of the continuous variables 5-132. Show that by expanding e in a power series and
X and Y is 0 < x < a and 0 < y < b. Also suppose that the joint taking expectations term by term we may write the moment-
probability density function f (x, y) = g(x)h(y), where g(x) generating function as
XY
t)
is a function only of x, and h(y) is a function only of y. Show M X ( = ( ) = + μ′ t + μ′ t 2 + r μ′ t r
tX
that X and Y are independent. E e 1 1 2 2! +⋯ r! +⋯
′
5-130. This exercise extends the hypergeometric distribu- Thus, the coeficient of / !t r r in this expansion is μ r , the rth
tion to multiple variables. Consider a population with N items origin moment.
of k different types. Assume that there are N items of type Write the power series expansion for M X ( t) for a
1
′
′
1, N items of type 2,…, N items of type k so that N + N + gamma random variable and determine μ 1 and μ 2 using this
2 k 1 2
… + … N = N. Suppose that a random sample of size n is
k approach.
Important Terms and Concepts
Bivariate distribution Contour plots Joint probability density Marginal probability
Bivariate normal distribution Correlation function distribution
Conditional mean Covariance Joint probability distribution Moment generating functions
Conditional probability Error propagation Joint probability mass Multinomial distribution
density function General functions of random function Reproductive property of the
Conditional probability mass variables Linear functions of random normal distribution
function Independence variables
Conditional variance