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$\newcommand\la\lambda$The random variables (r.v.'s) in question are of the form $X_{w,m,\la}:=wX_\la+m$, where $X_\la\sim\chi_1^2(\la)$.

First, trivial observations. Obviously, $X_{w,m,\la}$ is increasing in $m$ and hence stochastically increasing in $m$. Also, because $X_\la\ge0$, $X_{w,m,\la}$ is increasing in $w$ and hence stochastically increasing in $w$.

It is a bit less trivial that $X_\la$ is stochastically increasing in $\la$, and hence $X_{w,m,\la}$ is stochastically increasing in $\la$ for each pair $(w,m)$ such that $w\ge0$. One way to show this is to recall that the noncentral chi-squared distribution $\chi_1^2(\la)$ is the distribution of the r.v. $Y_{1+N_{\la/2}}$, where $Y_n$ has the central chi-squared with $n$ degrees of freedom (d.f.'s) and $N_{\la/2}$ is a r.v. with the Poisson distribution with mean $\la/2$. The precise meaning of the latter informal statement is that, for each nonnegative integer $k$, the conditional distribution of the r.v. $Y_{1+N_{\la/2}}$ given that $N_{\la/2}=k$ is the distribution of $Y_{1+k}$ -- that is, the central chi-squared distribution with $1+k$ d.f.'s. It remains to note the following:

• $Y_n$ is stochastically increasing in $n$, because $Y_n$ equals $Z_1^2+\cdots+Z_n^2$ in distribution, where $Z,Z_1,Z_2,\ldots$ are independent standard normal r.v.'s.
• A Poisson r.v. $N_{\mu}$ with mean $\mu$ is stochastically increasing in $\mu$, say because for any real $\mu$ and $\nu$ such that $0<\mu<\nu$, the r.v. $N_{\nu}$ equals $N'_{\mu}+N''_{\nu-\mu}$ in distribution (and $N''_{\nu-\mu}\ge0$), where $N'_{\mu}$ and $N''_{\nu-\mu}$ are independent Poisson r.v.'s with respective means $\mu$ and $\nu-\mu$.

A more direct (but more calculational) way to prove that that $X_\la$ is stochastically increasing in $\la$ is as follows. Without loss of generality, $$X_\la=(Z+\mu)^2,$$ where $\mu:=\sqrt\la$. It remains to note that, for each real $t\ge0$, $$p(\mu):=P(X_\la\le t^2)=P(\mu-t\le Z\le\mu+t)$$ and $p'(\mu)=g(\mu+t)-g(\mu-t)\le0$, where $g$ is the standard normal p.d.f.

$\newcommand\la\lambda$The random variables (r.v.'s) in question are of the form $X_{w,m,\la}:=wX_\la+m$, where $X_\la\sim\chi_1^2(\la)$.

First, trivial observations. Obviously, $X_{w,m,\la}$ is increasing in $m$ and hence stochastically increasing in $m$. Also, because $X_\la\ge0$, $X_{w,m,\la}$ is increasing in $w$ and hence stochastically increasing in $w$.

It is a bit less trivial that $X_\la$ is stochastically increasing in $\la$, and hence $X_{w,m,\la}$ is stochastically increasing in $\la$ for each pair $(w,m)$ such that $w\ge0$. One way to show this is to recall that the noncentral chi-squared distribution $\chi_1^2(\la)$ is the distribution of the r.v. $Y_{1+N_{\la/2}}$, where $Y_n$ has the central chi-squared with $n$ degrees of freedom (d.f.'s) and $N_{\la/2}$ is a r.v. with the Poisson distribution with mean $\la/2$. The precise meaning of the latter informal statement is that, for each nonnegative integer $k$, the conditional distribution of the r.v. $Y_{1+N_{\la/2}}$ given that $N_{\la/2}=k$ is the distribution of $Y_{1+k}$ -- that is, the central chi-squared distribution with $1+k$ d.f.'s. It remains to note the following:

• $Y_n$ is stochastically increasing in $n$, because $Y_n$ equals $Z_1^2+\cdots+Z_n^2$ in distribution, where $Z,Z_1,Z_2,\ldots$ are independent standard normal r.v.'s.
• A Poisson r.v. $N_{\mu}$ with mean $\mu$ is stochastically increasing in $\mu$, say because for any real $\mu$ and $\nu$ such that $0<\mu<\nu$, the r.v. $N_{\nu}$ equals $N'_{\mu}+N''_{\nu-\mu}$ in distribution (and $N''_{\nu-\mu}\ge0$), where $N'_{\mu}$ and $N''_{\nu-\mu}$ are independent Poisson r.v.'s with respective means $\mu$ and $\nu-\mu$.

A more direct (but more calculational) way to prove that that $X_\la$ is stochastically increasing in $\la$ is as follows. Without loss of generality (wlog), $$X_\la=(Z+\mu)^2,$$ where $\mu:=\sqrt\la$. It remains to note that, for each real $t\ge0$, $$p(\mu):=P(X_\la\le t^2)=P(\mu-t\le Z\le\mu+t)$$ and $p'(\mu)=g(\mu+t)-g(\mu-t)\le0$, where $g$ is the standard normal p.d.f.

It follows that, more generally, $X_{d,\la}$ is stochastically increasing in $\la$ for each natural $d$, where $X_{d,\la}$ is a r.v. with the noncentral chi-squared distribution $\chi_d^2(\la)$ with $d$ d.f.'s. Indeed, if $d\ge2$, then wlog $$X_{d,\la}=(Z_1+\mu)^2+Z_2^2+\cdots+Z_d^2,$$ again with $\mu:=\sqrt\la$.

$\newcommand\la\lambda$The random variables (r.v.'s) in question are of the form $X_{w,m,\la}:=wX_\la+m$, where $X_\la\sim\chi_1^2(\la)$.

First, trivial observations. Obviously, $X_{w,m,\la}$ is increasing in $m$ and hence stochastically increasing in $m$. Also, because $X_\la\ge0$, $X_{w,m,\la}$ is increasing in $w$ and hence stochastically increasing in $w$.

It is less trivial that $X_{w,m,\la}$ is stochastically increasing in $\la$ as well for each pair $(w,m)$ such that $w\ge0$. One way to show this is to recall that the noncentral chi-squared distribution $\chi_1^2(\la)$ is the distribution of the r.v. $Y_{1+N_{\la/2}}$, where $Y_n$ has the central chi-squared with $n$ degrees of freedom (d.f.'s) and $N_{\la/2}$ is a r.v. with the Poisson distribution with mean $\la/2$. The precise meaning of the latter informal statement is that, for each nonnegative integer $k$, the conditional distribution of the r.v. $Y_{1+N_{\la/2}}$ given that $N_{\la/2}=k$ is the distribution of $Y_{1+k}$ -- that is, the central chi-squared distribution with $1+k$ d.f.'s. It remains to note the following:

• $Y_n$ is stochastically increasing in $n$, because $Y_n$ equals $Z_1^2+\cdots+Z_n^2$ in distribution, where $Z_1,Z_2,\ldots$ are independent standard normal r.v.'s.
• A Poisson r.v. $N_{\mu}$ with mean $\mu$ is stochastically increasing in $\mu$, say because for any real $\mu$ and $\nu$ such that $0<\mu<\nu$, the r.v. $N_{\nu}$ equals $N'_{\mu}+N''_{\nu-\mu}$ in distribution (and $N''_{\nu-\mu}\ge0$), where $N'_{\mu}$ and $N''_{\nu-\mu}$ are independent Poisson r.v.'s with respective means $\mu$ and $\nu-\mu$.

$\newcommand\la\lambda$The random variables (r.v.'s) in question are of the form $X_{w,m,\la}:=wX_\la+m$, where $X_\la\sim\chi_1^2(\la)$.

First, trivial observations. Obviously, $X_{w,m,\la}$ is increasing in $m$ and hence stochastically increasing in $m$. Also, because $X_\la\ge0$, $X_{w,m,\la}$ is increasing in $w$ and hence stochastically increasing in $w$.

It is a bit less trivial that $X_\la$ is stochastically increasing in $\la$, and hence $X_{w,m,\la}$ is stochastically increasing in $\la$ for each pair $(w,m)$ such that $w\ge0$. One way to show this is to recall that the noncentral chi-squared distribution $\chi_1^2(\la)$ is the distribution of the r.v. $Y_{1+N_{\la/2}}$, where $Y_n$ has the central chi-squared with $n$ degrees of freedom (d.f.'s) and $N_{\la/2}$ is a r.v. with the Poisson distribution with mean $\la/2$. The precise meaning of the latter informal statement is that, for each nonnegative integer $k$, the conditional distribution of the r.v. $Y_{1+N_{\la/2}}$ given that $N_{\la/2}=k$ is the distribution of $Y_{1+k}$ -- that is, the central chi-squared distribution with $1+k$ d.f.'s. It remains to note the following:

• $Y_n$ is stochastically increasing in $n$, because $Y_n$ equals $Z_1^2+\cdots+Z_n^2$ in distribution, where $Z,Z_1,Z_2,\ldots$ are independent standard normal r.v.'s.
• A Poisson r.v. $N_{\mu}$ with mean $\mu$ is stochastically increasing in $\mu$, say because for any real $\mu$ and $\nu$ such that $0<\mu<\nu$, the r.v. $N_{\nu}$ equals $N'_{\mu}+N''_{\nu-\mu}$ in distribution (and $N''_{\nu-\mu}\ge0$), where $N'_{\mu}$ and $N''_{\nu-\mu}$ are independent Poisson r.v.'s with respective means $\mu$ and $\nu-\mu$.

A more direct (but more calculational) way to prove that that $X_\la$ is stochastically increasing in $\la$ is as follows. Without loss of generality, $$X_\la=(Z+\mu)^2,$$ where $\mu:=\sqrt\la$. It remains to note that, for each real $t\ge0$, $$p(\mu):=P(X_\la\le t^2)=P(\mu-t\le Z\le\mu+t)$$ and $p'(\mu)=g(\mu+t)-g(\mu-t)\le0$, where $g$ is the standard normal p.d.f.

$\newcommand\la\lambda$The random variables (r.v.'s) in question are of the form $X_{w,m,\la}:=wX_\la+m$, where $X_\la\sim\chi_1^2(\la)$.

First, trivial observations. Obviously, $X_{w,m,\la}$ is increasing in $m$ and hence stochastically increasing in $m$. Also, because $X_\la\ge0$, $X_{w,m,\la}$ is increasing in $w$ and hence stochastically increasing in $w$.

It is less trivial that $X_{w,m,\la}$ is stochastically increasing in $\la$ as well for each pair $(w,m)$ such that $w\ge0$. One way to show this is to recall that the noncentral chi-squared $\chi_1^2(\la)$ is the distribution of the r.v. $Y_{1+N_{\la/2}}$, where $Y_n$ has the central chi-squared with $n$ degrees of freedom (d.f.'s) and $N_{\la/2}$ is a r.v. with the Poisson distribution with mean $\la/2$. The precise meaning of the latter informal statement is that, for each nonnegative integer $k$, the conditional distribution of the r.v. $Y_{1+N_{\la/2}}$ given that $N_{\la/2}=k$ is the distribution of $Y_{1+k}$ -- that is, the central chi-squared distribution with $1+k$ d.f.'s. It remains to note the following:

• $Y_n$ is stochastically increasing in $n$, because $Y_n$ equals $Z_1^2+\cdots+Z_n^2$ in distribution, where $Z_1,Z_2,\ldots$ are independent standard normal r.v.'s.
• A Poisson r.v. $N_{\mu}$ with mean $\mu$ is stochastically increasing in $\mu$, say because for any real $\mu$ and $\nu$ such that $0<\mu<\nu$, the r.v. $N_{\nu}$ equals $N'_{\mu}+N''_{\nu-\mu}$ in distribution (and $N''_{\nu-\mu}\ge0$), where $N'_{\mu}$ and $N''_{\nu-\mu}$ are independent Poisson r.v.'s with respective means $\mu$ and $\nu-\mu$.

$\newcommand\la\lambda$The random variables (r.v.'s) in question are of the form $X_{w,m,\la}:=wX_\la+m$, where $X_\la\sim\chi_1^2(\la)$.

First, trivial observations. Obviously, $X_{w,m,\la}$ is increasing in $m$ and hence stochastically increasing in $m$. Also, because $X_\la\ge0$, $X_{w,m,\la}$ is increasing in $w$ and hence stochastically increasing in $w$.

It is less trivial that $X_{w,m,\la}$ is stochastically increasing in $\la$ as well for each pair $(w,m)$ such that $w\ge0$. One way to show this is to recall that the noncentral chi-squared distribution $\chi_1^2(\la)$ is the distribution of the r.v. $Y_{1+N_{\la/2}}$, where $Y_n$ has the central chi-squared with $n$ degrees of freedom (d.f.'s) and $N_{\la/2}$ is a r.v. with the Poisson distribution with mean $\la/2$. The precise meaning of the latter informal statement is that, for each nonnegative integer $k$, the conditional distribution of the r.v. $Y_{1+N_{\la/2}}$ given that $N_{\la/2}=k$ is the distribution of $Y_{1+k}$ -- that is, the central chi-squared distribution with $1+k$ d.f.'s. It remains to note the following:

• $Y_n$ is stochastically increasing in $n$, because $Y_n$ equals $Z_1^2+\cdots+Z_n^2$ in distribution, where $Z_1,Z_2,\ldots$ are independent standard normal r.v.'s.
• A Poisson r.v. $N_{\mu}$ with mean $\mu$ is stochastically increasing in $\mu$, say because for any real $\mu$ and $\nu$ such that $0<\mu<\nu$, the r.v. $N_{\nu}$ equals $N'_{\mu}+N''_{\nu-\mu}$ in distribution (and $N''_{\nu-\mu}\ge0$), where $N'_{\mu}$ and $N''_{\nu-\mu}$ are independent Poisson r.v.'s with respective means $\mu$ and $\nu-\mu$.