Answer:
a) Mean, E(X) = - 0.5
Variance = = 9.25
b)
or
⇒
⇒
Step-by-step explanation:
Given:
moment generating function of X as:
MX(t) =
a) Now
Mean, E(X) =
Thus,
or
also,
Thus,
or
Therefore,
Mean, E(X) =
or
Mean, E(X) = - 0.5
and
or
= 9.5
also,
Variance(X) = E(X²) - E(X)²
⇒ 9.5 - (-0.5)²
= 9.25
b) Now,
Let f(x) be the PMF of X
Thus,
or
⇒
⇒
Therefore,
at x = 0, P(x) =
at x= - 4 ,P(x) =
at x = 5, P(x) =
Thus,
E(X) =
or
E(X) = - 0.5
also,
= 9.5
Hence,
Var(X) = E(X²) - E(X)²
⇒ 9.5 - (-0.5)²
= 9.25