Suppose the MGF of X is
𝑀
𝑋
(
𝑠
)
=
(
0.2
+
0.8
𝑒
𝑠
)
7
. What distribution does X follow? (Use pattern matching with the summary box in the notes.)

Question

Suppose the MGF of X is 
𝑀
𝑋
(
𝑠
)
=
(
0.2
+
0.8
𝑒
𝑠
)
7
.  What distribution does X follow?  (Use pattern matching with the summary box in the notes.)

BlackTom AI · Accepted Answer

The moment generating function (MGF) for a random variable X provides a powerful tool for identifying the distribution of X. The MGF of a binomially distributed random variable X with parameters n and p is given by M_X(s) = (q + pe^s)^n, where q = 1 - p. In the given problem, the MGF is M_X(s) = (0.2 + 0.8e^s)^7.

Comparing this with the standard form, we identify q = 0.2 and p = 0.8, which implies that the random variable X follows a binomial distribution with n = 7 trials and success probability p = 0.8. The exponent 7 in the expression (0.2 + 0.8e^s)^7 directly corresponds to the number of trials n in the binomial distribution.

Now, let's analyze the incorrect options:

1. Bernoulli(p = 0.2): A Bernoulli distribution is a special case of the binomial distribution with n = 1. The given MGF has n = 7, not 1, so this option is incorrect.

2. Bernoulli(p = 0.8): Similar to the previous point, a Bernoulli distribution would require n = 1, which does not match the exponent 7 in the MGF.

3. Binomial(n = 7, p = 0.2): This option incorrectly assigns the success probability p as 0.2. However, the structure of the MGF clearly indicates that p = 0.8 (since 0.8 is the coefficient of e^s), making this option incorrect.

Therefore, the correct distribution that X follows is Binomial(n = 7, p = 0.8).

Suppose the MGF of X is 𝑀 𝑋 ( 𝑠 ) = ( 0.2 + 0.8 𝑒 𝑠 ) 7 . What distribution does X follow? (Use pattern matching with the summary box in the notes.)单项选择题

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