a die is tossed four times. A success is getting 1 or 2 on a toss find the mean and variance of the number of success
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Let x be the random variable denoting the number of times an odd number (the number of successes) when a die is tossed twice.
Then x takes the values 0,1,2
Let P(X=0) be probability of getting no odd number (both times showing even).
∴P(X=0)=
6
3
×
6
3
=
4
1
Let P(X=1) be probability of getting odd number once.
∴P(X=1)=2C
1
6
3
×
6
3
=
6
6
×
6
3
=
2
1
Let P(X=2) be probability of getting odd number twice.
∴P(X=2)=
6
3
×
6
3
=
4
1
Thus the probability distribution of X is given by
X=x x=0 x=1 x=2
P(X=x)
4
1
2
1
4
1
We know that mean E(X)=∑x
i
P
i
=0×
4
1
+1×
2
1
+2×
4
1
∴E(X)=0+
2
1
+
2
1
=1
Thus mean E(X)=1.
We know that var(X)=E(X
2
)−[E(X)]
2
E(X
2
)=∑x
i
2
P
i
=0×
4
1
+1
2
×
2
1
+2
2
×
4
1
∴E(X
2
)=0+
2
1
+4×
4
1
=
2
3
Thus var(X)=
2
3
−[1]
2
=
2
3
−1=
2
1
Hence mean is 1 and variance is
2
1
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