In the given Binomial distribution,
n=4andp=
1
3
andq=1-
1
3
=
2
3
Now,P(X=0)=4C0p0q4
1∗1∗(
2
3
)4=
16
81
P(X=1)=4C1p1q3
4∗
1
3
∗(
2
3
)3=
32
81
P(X=2)=4C2p2q2
6∗(
1
3
)2∗(
2
3
)2=
24
81
P(X=3)=4C3p3q1
4∗(
1
3
)3∗(
2
3
)3=
8
81
P(X=4)=4C4p4q0
1∗1∗(
1
3
)4=
1
81
Now, Mean =0∗
16
81
+1∗
32
81
+2∗
24
81
+3∗
8
81
+4∗
1
81
=
32
81
+
48
81
+
24
81
+
4
81
=
108
81
=
4
3
