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I have recently been given a set of practice problems for my probabilities course and I have no idea where to even start on this question.

The distribution of X = the number of toppings ordered by a randomly selected customer is given in the table below. It turns out that X is independent of the size of the pizza and the type of cheese and that each topping is equally popular as are the two cheese types.

P(X=x)

0 = 0.3

1 = 0.3

2 = 0.1

3 = 0.1

4 = 0.2

What is the mgf of X?

I am pretty sure this is a Binomial Distribution. So would I just put $[(1-θ) + θe^t]n$ as my answer?

Overall I am really confused on this topic and would like some help that isn't too discrete. Thank You!

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Dillon Hector
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    Please see our [help/on-topic], which has guidance on how to ask homework-like questions (whether or not this is actually homework). In particular you should attempt to ask about a specific problem you have with answering it yourself, and if you say what you think the answer is, it's important to explain why (how you got there) so we can help you understand why you went wrong. – Glen_b Sep 23 '18 at 04:22

1 Answers1

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$X$ is not a binomial distribution. A binomial distribution is the number of successes $X$ out of $n$ independent trials with constant probability of success $p$.

Now to answer your question, you're given the PMF of your random variable $X$ explicitly. The moment generating function of a random variable $X$ is defined as

$$M_X (t) = E[e^{Xt}]$$

And for a discrete random variable $Z$, expectation of $g(Z)$ is defined as

$E[g(Z)] = \sum_z g(z) \times P(Z=z)$

Therefore, the moment generating function of your random variable $X$ is defined as

$M_X(t) = \sum_x e^{xt} \times P(X=x) $

$= e^{0t}\times 0.3 + e^{1t}\times 0.3 + e^{2t}\times 0.1 + e^{3t}\times 0.1 + e^{4t} \times 0.2$

Which is clearly a function in $t$ as you would expect

  • So exactly what type of distribution would X be? If any. – Dillon Hector Sep 22 '18 at 14:28
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    It's not a specific type, it's described by the probability function given to you. So long as the function's probabilities sum to 1, it's a valid discrete distribution, it doesn't need to be of a specific type. –  Sep 22 '18 at 14:29
  • Okay, so say I did get a Binomial Distribution, would I use the formula and then that is it? Or would I do what was done above? Sorry just a bit confused. – Dillon Hector Sep 22 '18 at 14:31
  • Yes. If you know $X$ has a Binomial$(n,p)$ distribution, then you can use the formula for it's MGF rather than deriving it. In the case of parameters $n,p$, it would have MGF $(1-p+pe^t)^n$. You just need to make sure the binomial assumptions are actually true, or if you are explitly told in the question. No need to apologize, you're here to learn after all! –  Sep 22 '18 at 14:37
  • To further add: if you are explicitly told in the question "$X$ has ___ distribution with parameters ___" then you can just look up the distribution's wikipedia page and find it's MGF formula on the side bar. As long as you know the parameters, you can just chuck them in to the MGF formular right away –  Sep 22 '18 at 14:39
  • $X$ has a multinomial distribution. But you don't need to know that to follow @Xiaomi's clear reasoning here. – Frank Sep 22 '18 at 16:26
  • This is pretty clearly homework/self-study, though the OP, being new apparently hasn't read our guidelines for answering such questions -- consequently it's important that answerers be familiar with our standards. Please see our [help/on-topic] – Glen_b Sep 23 '18 at 04:19