Help on Moment Generating Functions
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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!
mgf
edited Sep 22 at 14:55
SecretAgentMan
405114
asked Sep 22 at 14:00
Dillon Hector
206
add a comment |Â
up vote
2
down vote
favorite
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!
mgf
edited Sep 22 at 14:55
SecretAgentMan
405114
asked Sep 22 at 14:00
Dillon Hector
206
Please see our help center, 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 at 4:22
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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!
mgf
edited Sep 22 at 14:55
SecretAgentMan
405114
asked Sep 22 at 14:00
Dillon Hector
206
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!
mgf
mgf
edited Sep 22 at 14:55
SecretAgentMan
405114
asked Sep 22 at 14:00
Dillon Hector
206
edited Sep 22 at 14:55
SecretAgentMan
405114
asked Sep 22 at 14:00
Dillon Hector
206
edited Sep 22 at 14:55
SecretAgentMan
405114
edited Sep 22 at 14:55
SecretAgentMan
405114
edited Sep 22 at 14:55
SecretAgentMan
405114
405114
asked Sep 22 at 14:00
Dillon Hector
206
asked Sep 22 at 14:00
Dillon Hector
206
asked Sep 22 at 14:00
Dillon Hector
206
206
Please see our help center, 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 at 4:22
add a comment |Â
Please see our help center, 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 at 4:22
Please see our help center, 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 at 4:22
Please see our help center, 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 at 4:22
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
8
down vote
accepted
$X$ 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) $
$= e^0ttimes 0.3 + e^1ttimes 0.3 + e^2ttimes 0.1 + e^3ttimes 0.1 + e^4t times 0.2$
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 at 14:28
1
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.
– user220802
Sep 22 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 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!
– user220802
Sep 22 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
– user220802
Sep 22 at 14:39
 |Â
show 2 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
8
down vote
accepted
$X$ 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) $
$= e^0ttimes 0.3 + e^1ttimes 0.3 + e^2ttimes 0.1 + e^3ttimes 0.1 + e^4t times 0.2$
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 at 14:28
1
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.
– user220802
Sep 22 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 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!
– user220802
Sep 22 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
– user220802
Sep 22 at 14:39
 |Â
show 2 more comments
up vote
8
down vote
accepted
$X$ 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) $
$= e^0ttimes 0.3 + e^1ttimes 0.3 + e^2ttimes 0.1 + e^3ttimes 0.1 + e^4t times 0.2$
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 at 14:28
1
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.
– user220802
Sep 22 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 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!
– user220802
Sep 22 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
– user220802
Sep 22 at 14:39
 |Â
show 2 more comments
up vote
8
down vote
accepted
up vote
8
down vote
accepted
$X$ 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) $
$= e^0ttimes 0.3 + e^1ttimes 0.3 + e^2ttimes 0.1 + e^3ttimes 0.1 + e^4t times 0.2$
Which is clearly a function in $t$ as you would expect
$X$ 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) $
$= e^0ttimes 0.3 + e^1ttimes 0.3 + e^2ttimes 0.1 + e^3ttimes 0.1 + e^4t times 0.2$
Which is clearly a function in $t$ as you would expect
edited Sep 22 at 16:03
answered Sep 22 at 14:26
user220802
So exactly what type of distribution would X be? If any.
– Dillon Hector
Sep 22 at 14:28
1
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.
– user220802
Sep 22 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 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!
– user220802
Sep 22 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
– user220802
Sep 22 at 14:39
 |Â
show 2 more comments
So exactly what type of distribution would X be? If any.
– Dillon Hector
Sep 22 at 14:28
1
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.
– user220802
Sep 22 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 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!
– user220802
Sep 22 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
– user220802
Sep 22 at 14:39
So exactly what type of distribution would X be? If any.
– Dillon Hector
Sep 22 at 14:28
So exactly what type of distribution would X be? If any.
– Dillon Hector
Sep 22 at 14:28
1
1
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.
– user220802
Sep 22 at 14:29
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.
– user220802
Sep 22 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 at 14:31
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 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!
– user220802
Sep 22 at 14:37
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!
– user220802
Sep 22 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
– user220802
Sep 22 at 14:39
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
– user220802
Sep 22 at 14:39
 |Â
show 2 more comments
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Please see our help center, 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 at 4:22