Let
c
0
,
c
1
,
c
2
,
⋯
{c_0},{c_1},{c_2}, \cdots
be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let
P
k
∗
(
x
)
(
k
=
0
,
1
,
2
,
⋯
)
P_k^ \ast (x)(k = 0,1,2, \cdots )
be the normalised Legendre polynomials orthogonal with respect to the interval
(
−
1
,
1
)
( - 1,1)
. It is proved that the average number of the zeros of
c
0
P
0
∗
(
x
)
+
c
1
P
1
∗
(
x
)
+
⋯
+
c
n
P
n
∗
(
x
)
{c_0}P_0^ \ast (x) + {c_1}P_1^ \ast (x) + \cdots + {c_n}P_n^ \ast (x)
in the same interval is asymptotically equal to
(
3
)
−
1
/
2
n
{(3)^{ - 1/2}}n
when
n
n
is large.