Compute $ Var(X+Y+Z) $ where $ X,Y,Z sim Binomial $
Suppose I throw 3 fair dice 30 times.
Let,
X = no' of throws in which we don't get 4
Y = no' of throws in which we get 4 in only one die (out of 3)
Z = no' of throws in which we get 4 in exactly two dice (out of 3)
Compute $ Var(X+Y+Z) $
by definition $ Var(X+Y+Z) = E[(X+Y+Z)^2] + (E[X+Y+Z])^2 $
$ E[X+Y+Z] = E[X] + E[Y] + E[Z] = 29.8611$ because each variable of a multinomial distribution is of binomial distribution. so I know $ (E[X+Y+Z])^2 = 29.8611^2 $ but how do I calculate $ E[(X+Y+Z)^2] $ ?
Suggestions please?
probability variance expected-value
add a comment |
Suppose I throw 3 fair dice 30 times.
Let,
X = no' of throws in which we don't get 4
Y = no' of throws in which we get 4 in only one die (out of 3)
Z = no' of throws in which we get 4 in exactly two dice (out of 3)
Compute $ Var(X+Y+Z) $
by definition $ Var(X+Y+Z) = E[(X+Y+Z)^2] + (E[X+Y+Z])^2 $
$ E[X+Y+Z] = E[X] + E[Y] + E[Z] = 29.8611$ because each variable of a multinomial distribution is of binomial distribution. so I know $ (E[X+Y+Z])^2 = 29.8611^2 $ but how do I calculate $ E[(X+Y+Z)^2] $ ?
Suggestions please?
probability variance expected-value
1
Consider the probability of getting three $4$s and then consider $Var(n-X-Y-Z)$
– Henry
Jan 3 at 21:11
1
Can I just set new variable $ W = $ no' of throws in which I get up to two 4's and then calculate the variance of $ W $ as it is binomial variable?
– bm1125
2 days ago
1
Yes - that works very well since each set of throws is independent and gives a yes/no answer to "up to two 4s"
– Henry
2 days ago
add a comment |
Suppose I throw 3 fair dice 30 times.
Let,
X = no' of throws in which we don't get 4
Y = no' of throws in which we get 4 in only one die (out of 3)
Z = no' of throws in which we get 4 in exactly two dice (out of 3)
Compute $ Var(X+Y+Z) $
by definition $ Var(X+Y+Z) = E[(X+Y+Z)^2] + (E[X+Y+Z])^2 $
$ E[X+Y+Z] = E[X] + E[Y] + E[Z] = 29.8611$ because each variable of a multinomial distribution is of binomial distribution. so I know $ (E[X+Y+Z])^2 = 29.8611^2 $ but how do I calculate $ E[(X+Y+Z)^2] $ ?
Suggestions please?
probability variance expected-value
Suppose I throw 3 fair dice 30 times.
Let,
X = no' of throws in which we don't get 4
Y = no' of throws in which we get 4 in only one die (out of 3)
Z = no' of throws in which we get 4 in exactly two dice (out of 3)
Compute $ Var(X+Y+Z) $
by definition $ Var(X+Y+Z) = E[(X+Y+Z)^2] + (E[X+Y+Z])^2 $
$ E[X+Y+Z] = E[X] + E[Y] + E[Z] = 29.8611$ because each variable of a multinomial distribution is of binomial distribution. so I know $ (E[X+Y+Z])^2 = 29.8611^2 $ but how do I calculate $ E[(X+Y+Z)^2] $ ?
Suggestions please?
probability variance expected-value
probability variance expected-value
asked Jan 3 at 20:56
bm1125
63516
63516
1
Consider the probability of getting three $4$s and then consider $Var(n-X-Y-Z)$
– Henry
Jan 3 at 21:11
1
Can I just set new variable $ W = $ no' of throws in which I get up to two 4's and then calculate the variance of $ W $ as it is binomial variable?
– bm1125
2 days ago
1
Yes - that works very well since each set of throws is independent and gives a yes/no answer to "up to two 4s"
– Henry
2 days ago
add a comment |
1
Consider the probability of getting three $4$s and then consider $Var(n-X-Y-Z)$
– Henry
Jan 3 at 21:11
1
Can I just set new variable $ W = $ no' of throws in which I get up to two 4's and then calculate the variance of $ W $ as it is binomial variable?
– bm1125
2 days ago
1
Yes - that works very well since each set of throws is independent and gives a yes/no answer to "up to two 4s"
– Henry
2 days ago
1
1
Consider the probability of getting three $4$s and then consider $Var(n-X-Y-Z)$
– Henry
Jan 3 at 21:11
Consider the probability of getting three $4$s and then consider $Var(n-X-Y-Z)$
– Henry
Jan 3 at 21:11
1
1
Can I just set new variable $ W = $ no' of throws in which I get up to two 4's and then calculate the variance of $ W $ as it is binomial variable?
– bm1125
2 days ago
Can I just set new variable $ W = $ no' of throws in which I get up to two 4's and then calculate the variance of $ W $ as it is binomial variable?
– bm1125
2 days ago
1
1
Yes - that works very well since each set of throws is independent and gives a yes/no answer to "up to two 4s"
– Henry
2 days ago
Yes - that works very well since each set of throws is independent and gives a yes/no answer to "up to two 4s"
– Henry
2 days ago
add a comment |
2 Answers
2
active
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These $X$, $Y$, and $Z$ are strongly related to each other - to the point where we'd rather not work with them directly in calculating the variance. In particular, $X+Y+Z+W=30$, where $W$ is the number of throws in which all three dice come up fours. The variance of $X+Y+Z$ is the same as that of $W$. Can you calculate that?
Thanks for the answer. But why is the variance of $ X+Y+Z $ be the same as that of $ W $ ?
– bm1125
2 days ago
Another poster got there while I was sleeping in more detail, but it's a general rule; the variance of $aX+b$ is $|a|^2$ times the variance of $X$ for constants $a$ and $b$. We write $X+Y+Z=30-W$ and apply that.
– jmerry
2 days ago
add a comment |
Using the notation from @jmerry, let $W$ be the number of tosses in which all 3 dice show 4. Thus, $X+Y+Z+W=30$, i.e. $X+Y+Z=30-W$ implying $Var(X+Y+Z)=Var(30-W)$.
Let's consider $Q=30-W$.
$mathbb{E}[Q] = 30-mathbb{E}[W]$
$mathbb{E}[Q^2] = 900 + mathbb{E}[W^2] - 60mathbb{E}[W]$
$Var[Q] = mathbb{E}[Q^2] - mathbb{E}[Q]^2$
$=900 + mathbb{E}[W^2] - 60mathbb{E}[W] - (30-mathbb{E}[W])^2$
$=mathbb{E}[W^2] -mathbb{E}[W]^2$
$=Var(W)$ (by definition)
Thus, $Var(30-W) = Var(W) = Var(X+Y+Z)$.
$W$ follows a binomial distribution with parameters $N=30$ and $p = left( frac{1}{6} right)^3$. You can calculate the variance as $Np(1-p)$.
add a comment |
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2 Answers
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2 Answers
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These $X$, $Y$, and $Z$ are strongly related to each other - to the point where we'd rather not work with them directly in calculating the variance. In particular, $X+Y+Z+W=30$, where $W$ is the number of throws in which all three dice come up fours. The variance of $X+Y+Z$ is the same as that of $W$. Can you calculate that?
Thanks for the answer. But why is the variance of $ X+Y+Z $ be the same as that of $ W $ ?
– bm1125
2 days ago
Another poster got there while I was sleeping in more detail, but it's a general rule; the variance of $aX+b$ is $|a|^2$ times the variance of $X$ for constants $a$ and $b$. We write $X+Y+Z=30-W$ and apply that.
– jmerry
2 days ago
add a comment |
These $X$, $Y$, and $Z$ are strongly related to each other - to the point where we'd rather not work with them directly in calculating the variance. In particular, $X+Y+Z+W=30$, where $W$ is the number of throws in which all three dice come up fours. The variance of $X+Y+Z$ is the same as that of $W$. Can you calculate that?
Thanks for the answer. But why is the variance of $ X+Y+Z $ be the same as that of $ W $ ?
– bm1125
2 days ago
Another poster got there while I was sleeping in more detail, but it's a general rule; the variance of $aX+b$ is $|a|^2$ times the variance of $X$ for constants $a$ and $b$. We write $X+Y+Z=30-W$ and apply that.
– jmerry
2 days ago
add a comment |
These $X$, $Y$, and $Z$ are strongly related to each other - to the point where we'd rather not work with them directly in calculating the variance. In particular, $X+Y+Z+W=30$, where $W$ is the number of throws in which all three dice come up fours. The variance of $X+Y+Z$ is the same as that of $W$. Can you calculate that?
These $X$, $Y$, and $Z$ are strongly related to each other - to the point where we'd rather not work with them directly in calculating the variance. In particular, $X+Y+Z+W=30$, where $W$ is the number of throws in which all three dice come up fours. The variance of $X+Y+Z$ is the same as that of $W$. Can you calculate that?
answered Jan 3 at 21:12
jmerry
2,276210
2,276210
Thanks for the answer. But why is the variance of $ X+Y+Z $ be the same as that of $ W $ ?
– bm1125
2 days ago
Another poster got there while I was sleeping in more detail, but it's a general rule; the variance of $aX+b$ is $|a|^2$ times the variance of $X$ for constants $a$ and $b$. We write $X+Y+Z=30-W$ and apply that.
– jmerry
2 days ago
add a comment |
Thanks for the answer. But why is the variance of $ X+Y+Z $ be the same as that of $ W $ ?
– bm1125
2 days ago
Another poster got there while I was sleeping in more detail, but it's a general rule; the variance of $aX+b$ is $|a|^2$ times the variance of $X$ for constants $a$ and $b$. We write $X+Y+Z=30-W$ and apply that.
– jmerry
2 days ago
Thanks for the answer. But why is the variance of $ X+Y+Z $ be the same as that of $ W $ ?
– bm1125
2 days ago
Thanks for the answer. But why is the variance of $ X+Y+Z $ be the same as that of $ W $ ?
– bm1125
2 days ago
Another poster got there while I was sleeping in more detail, but it's a general rule; the variance of $aX+b$ is $|a|^2$ times the variance of $X$ for constants $a$ and $b$. We write $X+Y+Z=30-W$ and apply that.
– jmerry
2 days ago
Another poster got there while I was sleeping in more detail, but it's a general rule; the variance of $aX+b$ is $|a|^2$ times the variance of $X$ for constants $a$ and $b$. We write $X+Y+Z=30-W$ and apply that.
– jmerry
2 days ago
add a comment |
Using the notation from @jmerry, let $W$ be the number of tosses in which all 3 dice show 4. Thus, $X+Y+Z+W=30$, i.e. $X+Y+Z=30-W$ implying $Var(X+Y+Z)=Var(30-W)$.
Let's consider $Q=30-W$.
$mathbb{E}[Q] = 30-mathbb{E}[W]$
$mathbb{E}[Q^2] = 900 + mathbb{E}[W^2] - 60mathbb{E}[W]$
$Var[Q] = mathbb{E}[Q^2] - mathbb{E}[Q]^2$
$=900 + mathbb{E}[W^2] - 60mathbb{E}[W] - (30-mathbb{E}[W])^2$
$=mathbb{E}[W^2] -mathbb{E}[W]^2$
$=Var(W)$ (by definition)
Thus, $Var(30-W) = Var(W) = Var(X+Y+Z)$.
$W$ follows a binomial distribution with parameters $N=30$ and $p = left( frac{1}{6} right)^3$. You can calculate the variance as $Np(1-p)$.
add a comment |
Using the notation from @jmerry, let $W$ be the number of tosses in which all 3 dice show 4. Thus, $X+Y+Z+W=30$, i.e. $X+Y+Z=30-W$ implying $Var(X+Y+Z)=Var(30-W)$.
Let's consider $Q=30-W$.
$mathbb{E}[Q] = 30-mathbb{E}[W]$
$mathbb{E}[Q^2] = 900 + mathbb{E}[W^2] - 60mathbb{E}[W]$
$Var[Q] = mathbb{E}[Q^2] - mathbb{E}[Q]^2$
$=900 + mathbb{E}[W^2] - 60mathbb{E}[W] - (30-mathbb{E}[W])^2$
$=mathbb{E}[W^2] -mathbb{E}[W]^2$
$=Var(W)$ (by definition)
Thus, $Var(30-W) = Var(W) = Var(X+Y+Z)$.
$W$ follows a binomial distribution with parameters $N=30$ and $p = left( frac{1}{6} right)^3$. You can calculate the variance as $Np(1-p)$.
add a comment |
Using the notation from @jmerry, let $W$ be the number of tosses in which all 3 dice show 4. Thus, $X+Y+Z+W=30$, i.e. $X+Y+Z=30-W$ implying $Var(X+Y+Z)=Var(30-W)$.
Let's consider $Q=30-W$.
$mathbb{E}[Q] = 30-mathbb{E}[W]$
$mathbb{E}[Q^2] = 900 + mathbb{E}[W^2] - 60mathbb{E}[W]$
$Var[Q] = mathbb{E}[Q^2] - mathbb{E}[Q]^2$
$=900 + mathbb{E}[W^2] - 60mathbb{E}[W] - (30-mathbb{E}[W])^2$
$=mathbb{E}[W^2] -mathbb{E}[W]^2$
$=Var(W)$ (by definition)
Thus, $Var(30-W) = Var(W) = Var(X+Y+Z)$.
$W$ follows a binomial distribution with parameters $N=30$ and $p = left( frac{1}{6} right)^3$. You can calculate the variance as $Np(1-p)$.
Using the notation from @jmerry, let $W$ be the number of tosses in which all 3 dice show 4. Thus, $X+Y+Z+W=30$, i.e. $X+Y+Z=30-W$ implying $Var(X+Y+Z)=Var(30-W)$.
Let's consider $Q=30-W$.
$mathbb{E}[Q] = 30-mathbb{E}[W]$
$mathbb{E}[Q^2] = 900 + mathbb{E}[W^2] - 60mathbb{E}[W]$
$Var[Q] = mathbb{E}[Q^2] - mathbb{E}[Q]^2$
$=900 + mathbb{E}[W^2] - 60mathbb{E}[W] - (30-mathbb{E}[W])^2$
$=mathbb{E}[W^2] -mathbb{E}[W]^2$
$=Var(W)$ (by definition)
Thus, $Var(30-W) = Var(W) = Var(X+Y+Z)$.
$W$ follows a binomial distribution with parameters $N=30$ and $p = left( frac{1}{6} right)^3$. You can calculate the variance as $Np(1-p)$.
answered 2 days ago
Aditya Dua
86418
86418
add a comment |
add a comment |
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1
Consider the probability of getting three $4$s and then consider $Var(n-X-Y-Z)$
– Henry
Jan 3 at 21:11
1
Can I just set new variable $ W = $ no' of throws in which I get up to two 4's and then calculate the variance of $ W $ as it is binomial variable?
– bm1125
2 days ago
1
Yes - that works very well since each set of throws is independent and gives a yes/no answer to "up to two 4s"
– Henry
2 days ago