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m 0 and mr3 0. Therefore the
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(a) The distribution has the appearance of Fig. 3-7. By symmetry, mr 1 coefficient of skewness is zero.
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CHAPTER 3 Mathematical Expectation
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(b) We have mr2 E(X2) 1 22p 2 2p 2 2p where we have made the transformation x2 > 2 (5) of Appendix A. Similarly we obtain mr 4 E(X4) 1 22p 4 2p 4 2p Now s2 m4 Thus the coefficient of kurtosis is m4 s4 3 E[(X E[(X
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1>2 30 v e
v dv
3 2
2 2p
v and used properties of the gamma function given in (2) and 2 22p
` 4 30 x e
1 1 2 2
4 3 `x e `
x2>2 dx
x2>2
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v dv
5 2
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m)2] m)4]
E(X )2 E(X4)
mr2 mr 4
1 2 1 3
3.42. Prove that
1 (see page 82).
For any real constant c, we have E[{Y Now the left side can be written E[(Y mY)2] c2E[(X mX)2] 2cE[(X mX)(Y mY)] s2 Y s2 Y s2 Y s2 c2 X s2 c2 X c2s2 X 2csXY 2csXY s2 X sXY 2 s2 X s2 c X mY c(X m)}2] 0
s2 s2 s2 X Y XY 2 sX
sXY 2 s2 X
s2 XY s2 X
CHAPTER 3 Mathematical Expectation
In order for this last quantity to be greater than or equal to zero for every value of c, we must have s2 s2 X Y which is equivalent to r2 1 or 1 r 1. s2 Y X 0 or s2 Y X s2 s2 X Y 1
SUPPLEMENTARY PROBLEMS
Expectation of random variables
3.43. A random variable X is defined by X 2 3 1 prob. 1>3 prob. 1>2. prob. 1>6 Find (a) E(X ), (b) E(2X e 5), (c) E(X2).
3.44. Let X be a random variable defined by the density function f (x) Find (a) E(X), (b) E(3X 2), (c) E(X2). e e 0
3x2 0
0 x 1 . otherwise
3.45. The density function of a random variable X is f (x) Find (a) E(X), (b) E(X2), (c) E[(X 1)2].
x 0 . otherwise
3.46. What is the expected number of points that will come up in 3 successive tosses of a fair die Does your answer seem reasonable Explain. e e 0
3.47. A random variable X has the density function f (x)
0 . Find E(e2X>3). 0
3.48. Let X and Y be independent random variables each having density function f (u) Find (a) E(X 3.49. Does (a) E(X Y), (b) E(X2 Y) E(X ) Y2), (c) E(XY ). E(Y), (b) E(XY ) E(X)E(Y), in Problem 3.48 Explain. e 2e 0
u 0 otherwise
3.50. Let X and Y be random variables having joint density function f (x, y) Find (a) E(X), (b) E(Y ), (c) E(X 3.51. Does (a) E(X Y) E(X )
3 x(x e5 0
0 x 1, 0 otherwise
Y), (d) E(XY ). E(X)E(Y), in Problem 3.50 Explain.
E(Y), (b) E(XY )
3.52. Let X and Y be random variables having joint density f (x, y) Find (a) E(X), (b) E(Y ), (c) E(X e 4xy 0 0 x 1, 0 otherwise y 1
Y), (d) E(XY ).
CHAPTER 3 Mathematical Expectation
3.53. Does (a) E(X
1 (2x e4 0
E(X) y)
E(Y), (b) E(XY ) 0 x 1, 0 otherwise y
E(X ) E(Y ), in Problem 3.52 Explain. 2
3.54. Let f (x, y) (e) E(X
. Find (a) E(X ), (b) E(Y ), (c) E(X2), (d) E(Y2),
Y), (f) E(XY).
3.55. Let X and Y be independent random variables such that X Find (a) E(3X 2Y ), (b) E(2X2 e 1 0 prob. 1>3 prob. 2>3 Y e 2 3 prob. 3>4 prob. 1>4
Y2), (c) E(XY ), (d) E(X2Y ).
3.56. Let X1, X2, . . . , Xn be n random variables which are identically distributed such that Xk c 1 2 1 prob. 1>2 prob. 1>3 prob. 1>6
Find (a) E(Xl
Xn ), (b) E(X2 1
X2 2
X2). n
Variance and standard deviation
3.57. Find (a) the variance, (b) the standard deviation of the number of points that will come up on a single toss of a fair die. 3.58. Let X be a random variable having density function f (x) Find (a) Var(X ), (b) sX. 3.59. Let X be a random variable having density function f (x) Find (a) Var(X ), (b) sX. 3.60. Find the variance and standard deviation for the random variable X of (a) Problem 3.43, (b) Problem 3.44. 3.61. A random variable X has E(X ) 2, E(X2) 8. Find (a) Var(X ), (b) sX. 1)2] 10, E[(X 2)2] 6 find (a) E(X ), (b) Var(X ), (c) sX. e e 0
1>4 0
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