Question Type 2: Using n and p as parameters and finding what values of either satisfy some conditions Bootcamps

For a binomial random variable $X\sim \text{Bin}(n,p)$, find expressions for the mean $\mu$ and variance $\sigma^2$ in terms of $n$ and $p$.

Let $X\sim \text{Bin}(10,p)$. Find the value of $p$ such that the mean of $X$ is 3.

Suppose $X\sim \text{Bin}(n,0.2)$. Find the smallest integer $n$ such that the variance of $X$ is at least 4.

For $X\sim \text{Bin}(n,p)$, find $p$ such that the mean equals the variance.

Find $p$ in terms of $n$ for $X\sim \text{Bin}(n,p)$ such that the variance is one quarter of the mean.

Find $p$ in terms of $n$ for $X\sim \text{Bin}(n,p)$ so that the mean of $X$ is twice its variance.

If $X\sim \text{Bin}(n,p)$ satisfies $P(X=0)=0.1$, express $n$ in terms of $p$.

Find $p$ for $X\sim \text{Bin}(n,p)$ so that the coefficient of variation $\mathrm{CV}=1$, where $\mathrm{CV}=\sigma/\mu$.

For $X\sim \text{Bin}(20,p)$, find $p$ such that $P(X\ge1)=0.95$.

Given $X\sim \text{Bin}(n,p)$ with mean $5$ and variance $4$, find the values of $n$ and $p$.

Find integer $n>0$ and rational $p$ such that for $X\sim \text{Bin}(n,p)$ the mean is $7.5$ and the variance is $3.75$.

For $X\sim \text{Bin}(n,p)$, find $p$ such that the excess kurtosis equals $\frac{1-6p(1-p)}{np(1-p)}=1$.