Rit (Vísindafélag Íslendinga) - 01.06.1931, Blaðsíða 11

11
_ (x-*.)1
or y = Ce 21, (17)
which is the normal or Qaussian frequency curve.
B. Let = a3 = a4 = & = /?3 = /94 = 0.
The relations (16a—e) give:
ao+A — 0, ?.2ai+/?o — 0, A3ai-f-2A2/?i — 0,
Xiai~\-3XaPi = 0, X*,ai~\-QXifii = 0.
Hence
fti----«o
ai_ __
ao
dy =
Y
ai
2X2
A3
2L> 3X3 4A.4
A3 A4 ^.5
2/12 (x — Xi) + A3
c. Ao o
«o A>=-
dx
or
y = Ce
A3 (x — Xi) + 2X2
2XS\ J+
(x—/.1+ -j—) h'
Ao
2a2-
J.3
(18)
This represents frequency curves of Pearson’s type III.
We learn from these examples that the higher semi-invari-
ants have to satisfy certain conditions, that for the Gaussian
curve A3=A.4=A.5=... = 0, but for the Pearson’s type III
2^*2 3^s 4+
h h h
C. Let «2 = a3 = a4 = /?3 = ^4 = 0 and we get, from
(16a—e), the following equations:
ao+ft = 0, +ai+/?o+3A2/?2 = 0,
^3ai+2^2^i+4^3/ít2 = 0, ^•4ai+3^3/í’i+(6^22+5+)/?2 = 0,
+ai+4^4^i+(24+/i.3+6+)/?2 = 0.
Substituting the values of ao, ai, /50 and + from the first
four equations, the differential equation becomes:
1 dy ____
y dx
(12+3+10+^4-----12^32)(x - ^l)+A3(6^-22 + ^4)______
(3A3a-2A2Z4) (x—Aj)2—^3(6^22+^i) (x-+)—A2(l2^2S+4^-2^4—3^-32)
(19)