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PSLE Foundation Mathematics

<p>The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where&nbsp;$n\geq2$ and $0&lt;p&lt;1$. The mean of $X$ is three times its standard deviation.</p>
<ol class="abc-list"><li data-list-item-id="bimodal-mean-sd-q2-a" data-list-item-path="bimodal-mean-sd-q2-a"><div class="list-item-paragraph">Show that&nbsp;$p=\frac{9}{n+9}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="bimodal-mean-sd-q2-b" data-list-item-path="bimodal-mean-sd-q2-b"><div class="list-item-paragraph">Given that&nbsp;$X$ is bimodal and $n&lt;40$, find all possible values of $n$ and the corresponding modes of $X$.<span class="inline-marks">[4]</span></div></li></ol>

The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n\geq2$ and $0<p<1$. The mean of $X$ is three times its standard deviation.

Determining an Unknown Binomial Success Probability

Bimodal Binomial Distributions

Mean and Variance of a Binomial Distribution

Two modes under a mean-standard-deviation condition

Discrete Random Variables (DRV)

<p>The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where&nbsp;$n$ is a positive integer and $0&lt;p&lt;1$. Denote $\mathrm{P}(X=r)$ by $q_r$.</p>
<ol class="abc-list"><li data-list-item-id="bimodal-mean-q1-a" data-list-item-path="bimodal-mean-q1-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{q_r}{q_{r-1}}=\frac{(n-r+1)p}{r(1-p)}$, for&nbsp;$r=1,2,\ldots,n$. Hence show that&nbsp;$X$ is bimodal if and only if&nbsp;$(n+1)p$ is an integer, and state its two modes in terms of $n$ and $p$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="bimodal-mean-q1-b" data-list-item-path="bimodal-mean-q1-b"><div class="list-item-paragraph">It is known that the modes of $X$ are $11$ and $12$, and that&nbsp;$\mathrm{E}(X)=11.52$. Determine $n$ and $p$.<span class="inline-marks">[3]</span></div></li></ol>

The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n$ is a positive integer and $0<p<1$. Denote $\mathrm{P}(X=r)$ by $q_r$.

Binomial Probabilities Using the Probability Mass Function

Two modes from a probability ratio and mean

<p>The random variable $X$ has distribution $\mathrm{B}\left( n,p \right)$, where&nbsp;$n\ge 2$. It is known that the mean of $X$ and the standard deviation of $X$ are equal.</p>
<ol class="abc-list"><li data-list-item-id="item-1760845825629-mf2jn5n3h" data-list-item-path="item-1760845825629-mf2jn5n3h"><div class="list-item-paragraph">Find an expression for&nbsp;$p$ in terms of $n$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760845827122-fg268dk4z" data-list-item-path="item-1760845827122-fg268dk4z"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{P}\left( X=1 \right)}{\mathrm{P}\left( X=2 \right)}=\frac{2n}{n-1}$. Hence find the the mode(s) of $X$.<span class="inline-marks">[4]</span></div></li></ol>

The random variable $X$ has distribution $\mathrm{B}\left( n,p \right)$, where $n\ge 2$. It is known that the mean of $X$ and the standard deviation of $X$ are equal.

Mode of a Binomial Distribution

2024 EJC P2 Q6

<p>The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$.</p>
<p>Denoting $\mathrm{P}\left( X=k \right)$ by ${{p}_{k}}$, show that&nbsp;$\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{\left( n-k+1 \right)p}{k\left( 1-p \right)},k=1,2,...,n$.</p>
<p>A distribution is said to be bimodal if it has two modes.</p>
<p>Find the $3$ least values of $n$, and the corresponding modes of $X$, given that&nbsp;$X$ is bimodal and that&nbsp;$p=\frac{9}{16}$.</p>

The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$.

Denoting $\mathrm{P}\left( X=k \right)$ by ${{p}_{k}}$, show that $\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{\left( n-k+1 \right)p}{k\left( 1-p \right)},k=1,2,...,n$.

A distribution is said to be bimodal if it has two modes.

Find the $3$ least values of $n$, and the corresponding modes of $X$, given that $X$ is bimodal and that $p=\frac{9}{16}$.

Prelims Revision Set: Bimodal Distribution Q2

<p>The random variable $X$ follows a binomial distribution $\mathrm{B}\left( 15,p \right)$.</p>
<p>Given that $X$ has two modes, $X=5$ and $X=6$, find the exact value of $p$.</p>

The random variable $X$ follows a binomial distribution $\mathrm{B}\left( 15,p \right)$.

Given that $X$ has two modes, $X=5$ and $X=6$, find the exact value of $p$.

Prelims Revision Set: Bimodal Distribution Q1

<ol class="abc-list"><li><div class="list-item-paragraph">The random variable $X$ follows a binomial distribution $\mathrm{B}\left( 10,p \right)$.</div><ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$X$ has two modes, $X=4$ and $X=5$, find the exact value of $p$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given instead that&nbsp;$\mathrm{P}\left( X\le 9 \right)=\frac{1023}{1024}$, find the exact value of $p$.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The random variable $Y$ follows a binomial distribution $\mathrm{B}\left( 500,0.5 \right)$</div><div class="list-item-paragraph">A sample of $30$ independent values of $Y$is recorded.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the probability that all the values recorded are less than or equal to $256$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">The mean of the $30$ values is calculated. Estimate the probability that this sample mean is less than or equal to $256$, stating clearly the approximation used.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Explain why the probability found in part <strong>(ii)</strong> is larger than that found in part <strong>(i)</strong>.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Central Limit Theorem

Binomial Probabilities Using Technology

Sample Mean and Its Sampling Distribution

2017 CJC P2 Q7

Sampling

<p>The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$.</p>
<p>Denoting $\mathrm{P}(X=k)$ by ${{p}_{k}}$, it is given that $\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{(n-k+1)p}{k(1-p)}$, $k=1$, $2$, …, $n$. </p>
<p>A distribution is said to be bimodal if it has two modes.</p>
<p>Find the least value of $n$, and the corresponding modes of $X$, given that $X$ is bimodal and that $p=\frac{18}{25}$.</p>

Denoting $\mathrm{P}(X=k)$ by ${{p}_{k}}$, it is given that $\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{(n-k+1)p}{k(1-p)}$, $k=1$, $2$, …, $n$. 

Find the least value of $n$, and the corresponding modes of $X$, given that $X$ is bimodal and that $p=\frac{18}{25}$.

2018 ACJC MYE Q8

<p>The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$.</p>
<p>Denoting $\mathrm{P}\left( X=k \right)$ by ${{p}_{k}}$, it is given that&nbsp;$\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{\left( n-k+1 \right)p}{k\left( 1-p \right)},k=1,2,...,n$.</p>
<p>A distribution is said to be bimodal if it has two modes.</p>
<p>Find the least value of $n$, and the corresponding modes of $X$, given that&nbsp;$X$ is bimodal and that&nbsp;$p=\frac{18}{25}$.<span class="inline-marks">[3]</span></p>

Denoting $\mathrm{P}\left( X=k \right)$ by ${{p}_{k}}$, it is given that $\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{\left( n-k+1 \right)p}{k\left( 1-p \right)},k=1,2,...,n$.

2017 ACJC MYE P2 Q8

<p>The random variable $X$ has a binomial distribution $\mathrm{B}\left( n,p \right)$, where&nbsp;$0 &lt; p &lt; 1$, and $n$ is an integer. Show that</p>
<p style="text-align: center">$\frac{\mathrm{P}\left( X=r \right)}{\mathrm{P}\left( X=r-1 \right)}=\left( \frac{n-r+1}{r} \right)\left( \frac{p}{1-p} \right)$.<span class="inline-marks">[3]</span></p>
<p>Hence find a condition relating $n$ and $p$ such that&nbsp;$X$ has two values for its mode, and determine these two values, giving your answer in terms of $n$ and $p$.<span class="inline-marks">[3]</span></p>

The random variable $X$ has a binomial distribution $\mathrm{B}\left( n,p \right)$, where $0 < p < 1$, and $n$ is an integer. Show that

$\frac{\mathrm{P}\left( X=r \right)}{\mathrm{P}\left( X=r-1 \right)}=\left( \frac{n-r+1}{r} \right)\left( \frac{p}{1-p} \right)$.

Hence find a condition relating $n$ and $p$ such that $X$ has two values for its mode, and determine these two values, giving your answer in terms of $n$ and $p$.

Math Question Bank