To find the mean of the given probability distribution, we use the formula for the expected value of a discrete random variable. The expected value (mean) of X is given by: E(X)=
∑
i
xiâ‹…P(X=xi) Given the values: X=xi:2,3,4,5
and corresponding probabilities: P(X=xi):‌
5
k
,‌
7
k
,‌
9
k
,‌
11
k
First, we need to find the value of the constant k to ensure that the sum of the probabilities equals 1 : ‌
5
k
+‌
7
k
+‌
9
k
+‌
11
k
=1
Simplifying, we get: ‌
5+7+9+11
k
=1⟹‌
32
k
=1⟹k=32 With k=32, the probability values become:
P(X=2)=‌
5
32
,‌‌P(X=3)=‌
7
32
,‌‌P(X=4)=‌
9
32
,‌‌P(X=5)=‌
11
32
Now we calculate the mean (expected value) E(X) : ‌E(X)=2⋅‌
5
32
+3⋅‌
7
32
+4⋅‌
9
32
+5⋅‌
11
32
‌E(X)=‌
10
32
+‌
21
32
+‌
36
32
+‌
55
32
‌E(X)=‌
10+21+36+55
32
‌E(X)=‌
122
32
‌E(X)=‌
61
16
So, the mean of this distribution is Option A: ‌