Question

the symmetric distribution having parameters as it's mean and variance is called as

Answer 1

Let
X
have a uniform distribution on (,)
(
a
,
b
)
. The density function of
X
is

()=1−
f
(
x
)
=
1
b
−
a
if ≤≤
a
≤
x
≤
b
and 0
0
elsewhere

The the mean is given by

[]=∫−=2−22(−)=+2
E
[
X
]
=
∫
a
b
x
b
−
a
d
x
=
b
2
−
a
2
2
(
b
−
a
)
=
b
+
a
2

The variance is given by[2]−([])2
E
[
X
2
]
−
(
E
[
X
]
)
2

[2]=∫2−=3−33(−)=2++23
E
[
X
2
]
=
∫
a
b
x
2
b
−
a
d
x
=
b
3
−
a
3
3
(
b
−
a
)
=
b
2
+
b
a
+
a
2
3

The required variance is then

2++23−(+)24=(−)212
b
2
+
b
a
+
a
2
3
−
(
b
+
a
)
2
4
=
(
b
−
a
)
2
12
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1 comment from Dauren Baitursyn
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Hope this helps

Answer 2

In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function or probability mass function is reflected around a vertical line at some value of the random variable represented by the distribution. This vertical line is the line of symmetry of the distribution. Thus the probability of being any given distance on one side of the value about which symmetry occurs is the same as the probability of being the same distance on the other side of that value.