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Find $mathrm E (X), mathrm Var (X)$.
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A sample of size $n$ is drawn from $1,2, cdots , N $ with replacement. Let $X$ denote the minimum of the numbers drawn. Calculate
$(a)$ the PMF of $X,$
$(b)$ $mathrm E (X),$
$(c)$ $mathrm Var (X),$
$(d)$ If $Y$ denotes the maximum of the numbers drawn$,$ calculate the joint PMF of $X$ and $Y$.
I have found
$$P(X=k)= frac (N-k+1)^n - (N-k)^n N^n.$$
But this isn't much helpful in calculating the expectation and variance of the random variable $X$. So how should I proceed in this regard? Please help me.
Thank you very much.
I have found the joint probability of the random variables $X$ and $Y$ as follows $:$
$$P(X=i,Y=j) = frac (j-i+1)^n - 2(j-i)^n+(j-i-1)^n N^n.$$ for $i=1,2,cdots,N$ and $j=1,2,,cdots,N$.
Is it correct? Please verify it.
probability probability-theory expectation variance
asked yesterday
Debabrata Chattopadhyay.
8311
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up vote
2
down vote
favorite
A sample of size $n$ is drawn from $1,2, cdots , N $ with replacement. Let $X$ denote the minimum of the numbers drawn. Calculate
$(a)$ the PMF of $X,$
$(b)$ $mathrm E (X),$
$(c)$ $mathrm Var (X),$
$(d)$ If $Y$ denotes the maximum of the numbers drawn$,$ calculate the joint PMF of $X$ and $Y$.
I have found
$$P(X=k)= frac (N-k+1)^n - (N-k)^n N^n.$$
But this isn't much helpful in calculating the expectation and variance of the random variable $X$. So how should I proceed in this regard? Please help me.
Thank you very much.
I have found the joint probability of the random variables $X$ and $Y$ as follows $:$
$$P(X=i,Y=j) = frac (j-i+1)^n - 2(j-i)^n+(j-i-1)^n N^n.$$ for $i=1,2,cdots,N$ and $j=1,2,,cdots,N$.
Is it correct? Please verify it.
probability probability-theory expectation variance
asked yesterday
Debabrata Chattopadhyay.
8311
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
A sample of size $n$ is drawn from $1,2, cdots , N $ with replacement. Let $X$ denote the minimum of the numbers drawn. Calculate
$(a)$ the PMF of $X,$
$(b)$ $mathrm E (X),$
$(c)$ $mathrm Var (X),$
$(d)$ If $Y$ denotes the maximum of the numbers drawn$,$ calculate the joint PMF of $X$ and $Y$.
I have found
$$P(X=k)= frac (N-k+1)^n - (N-k)^n N^n.$$
But this isn't much helpful in calculating the expectation and variance of the random variable $X$. So how should I proceed in this regard? Please help me.
Thank you very much.
I have found the joint probability of the random variables $X$ and $Y$ as follows $:$
$$P(X=i,Y=j) = frac (j-i+1)^n - 2(j-i)^n+(j-i-1)^n N^n.$$ for $i=1,2,cdots,N$ and $j=1,2,,cdots,N$.
Is it correct? Please verify it.
probability probability-theory expectation variance
asked yesterday
Debabrata Chattopadhyay.
8311
A sample of size $n$ is drawn from $1,2, cdots , N $ with replacement. Let $X$ denote the minimum of the numbers drawn. Calculate
$(a)$ the PMF of $X,$
$(b)$ $mathrm E (X),$
$(c)$ $mathrm Var (X),$
$(d)$ If $Y$ denotes the maximum of the numbers drawn$,$ calculate the joint PMF of $X$ and $Y$.
I have found
$$P(X=k)= frac (N-k+1)^n - (N-k)^n N^n.$$
But this isn't much helpful in calculating the expectation and variance of the random variable $X$. So how should I proceed in this regard? Please help me.
Thank you very much.
I have found the joint probability of the random variables $X$ and $Y$ as follows $:$
$$P(X=i,Y=j) = frac (j-i+1)^n - 2(j-i)^n+(j-i-1)^n N^n.$$ for $i=1,2,cdots,N$ and $j=1,2,,cdots,N$.
Is it correct? Please verify it.
probability probability-theory expectation variance
asked yesterday
Debabrata Chattopadhyay.
8311
edited yesterday
asked yesterday
Debabrata Chattopadhyay.
8311
asked yesterday
Debabrata Chattopadhyay.
8311
asked yesterday
Debabrata Chattopadhyay.
8311
8311
add a comment |Â
add a comment |Â
1 Answer
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Your answer for part (a) is correct.
For part (b), write your result from part (a) as
$$mathsf P(Xgt k)=frac(N-k)^nN^n$$
and calculate
$$
mathsf E(X)=sum_k=0^N-1mathsf P(Xgt k)=sum_k=1^Nfrack^nN^n;,
$$
which can be evaluated using Faulhaber's formula. Approximating the sum by an integral for large $N$ yields the approximation $E(X)approxfrac Nn+1$.
For part (c), use the fact that the variance of the minimum is the variance of the maximum to transform to $j=N-k+1$, which leads to $j^n-(j-1)^n$ in the numerator; then use $j=(j-1)+1$ as appropriate to write the expectation of $j^2$ in terms of sums of powers of $j$.
Your answer for part (d) is correct; it was recently asked about in
Calculating probability of High+Low = T on N dice with S sides.
answered yesterday
joriki
164k10179328
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Your answer for part (a) is correct.
For part (b), write your result from part (a) as
$$mathsf P(Xgt k)=frac(N-k)^nN^n$$
and calculate
$$
mathsf E(X)=sum_k=0^N-1mathsf P(Xgt k)=sum_k=1^Nfrack^nN^n;,
$$
which can be evaluated using Faulhaber's formula. Approximating the sum by an integral for large $N$ yields the approximation $E(X)approxfrac Nn+1$.
For part (c), use the fact that the variance of the minimum is the variance of the maximum to transform to $j=N-k+1$, which leads to $j^n-(j-1)^n$ in the numerator; then use $j=(j-1)+1$ as appropriate to write the expectation of $j^2$ in terms of sums of powers of $j$.
Your answer for part (d) is correct; it was recently asked about in
Calculating probability of High+Low = T on N dice with S sides.
answered yesterday
joriki
164k10179328
add a comment |Â
up vote
1
down vote
Your answer for part (a) is correct.
For part (b), write your result from part (a) as
$$mathsf P(Xgt k)=frac(N-k)^nN^n$$
and calculate
$$
mathsf E(X)=sum_k=0^N-1mathsf P(Xgt k)=sum_k=1^Nfrack^nN^n;,
$$
which can be evaluated using Faulhaber's formula. Approximating the sum by an integral for large $N$ yields the approximation $E(X)approxfrac Nn+1$.
For part (c), use the fact that the variance of the minimum is the variance of the maximum to transform to $j=N-k+1$, which leads to $j^n-(j-1)^n$ in the numerator; then use $j=(j-1)+1$ as appropriate to write the expectation of $j^2$ in terms of sums of powers of $j$.
Your answer for part (d) is correct; it was recently asked about in
Calculating probability of High+Low = T on N dice with S sides.
answered yesterday
joriki
164k10179328
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Your answer for part (a) is correct.
For part (b), write your result from part (a) as
$$mathsf P(Xgt k)=frac(N-k)^nN^n$$
and calculate
$$
mathsf E(X)=sum_k=0^N-1mathsf P(Xgt k)=sum_k=1^Nfrack^nN^n;,
$$
which can be evaluated using Faulhaber's formula. Approximating the sum by an integral for large $N$ yields the approximation $E(X)approxfrac Nn+1$.
For part (c), use the fact that the variance of the minimum is the variance of the maximum to transform to $j=N-k+1$, which leads to $j^n-(j-1)^n$ in the numerator; then use $j=(j-1)+1$ as appropriate to write the expectation of $j^2$ in terms of sums of powers of $j$.
Your answer for part (d) is correct; it was recently asked about in
Calculating probability of High+Low = T on N dice with S sides.
answered yesterday
joriki
164k10179328
Your answer for part (a) is correct.
For part (b), write your result from part (a) as
$$mathsf P(Xgt k)=frac(N-k)^nN^n$$
and calculate
$$
mathsf E(X)=sum_k=0^N-1mathsf P(Xgt k)=sum_k=1^Nfrack^nN^n;,
$$
which can be evaluated using Faulhaber's formula. Approximating the sum by an integral for large $N$ yields the approximation $E(X)approxfrac Nn+1$.
For part (c), use the fact that the variance of the minimum is the variance of the maximum to transform to $j=N-k+1$, which leads to $j^n-(j-1)^n$ in the numerator; then use $j=(j-1)+1$ as appropriate to write the expectation of $j^2$ in terms of sums of powers of $j$.
Your answer for part (d) is correct; it was recently asked about in
Calculating probability of High+Low = T on N dice with S sides.
answered yesterday
joriki
164k10179328
answered yesterday
joriki
164k10179328
answered yesterday
joriki
164k10179328
answered yesterday
joriki
164k10179328
164k10179328
add a comment |Â
add a comment |Â
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