The random variable $X$ has distribution $\mathrm{B}\left( n,p \right)$. It is given that the me

Question

The random variable $X$ has distribution $\mathrm{B}\left( n,p \right)$. It is given that the mean of $X$ is $4.5$ and the variance of $X$ is $3.15$. Find the values of $n$ and $p$.[3] Using the values of $n$ and $p$ found in part (i), find $\mathrm{P}\left( X\ge 5 \right)$.[2] Do not use a calculator in answering this part. It is given instead that the mode of $X$ is $10$ and $p=\frac{1}{2}$. By considering $\mathrm{P}\left( X=10 \right)>\mathrm{P}\left( X=9 \right)$ and $\mathrm{P}\left( X=10 \right)>\mathrm{P}\left( X=11 \right)$, write down an inequality involving $n$ and find the value of $n$.[4]

Accepted Answer

(i) $n = 15, \, p = \frac{3}{10}$ (ii) $\mathrm{P}(X \geq 5) = 0.485$ (iv) $n = 20$

2022 CJC P2 Q10

2022 CJC P2 Q10

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