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<title>Distributions in R</title>
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<h1 class="title toc-ignore">Distributions in R</h1>
<h4 class="date">21.11.2024</h4>
</div>
<ol style="list-style-type: decimal">
<li>In addition to the <code>d</code> and <code>p</code> functions
mentioned in class, R includes <code>q</code> and <code>r</code>
functions. What do these do? Explore the help files.</li>
</ol>
<div class="soln">
<p><code>q</code>: quantiles; returns the inverse of <code>p</code>.</p>
<ul>
<li>Example: <code>qnorm(0.2)</code> gives you the value <span
class="math inline">\(x\)</span> such that 20% of the values of a
standard normal are <= <span class="math inline">\(x\)</span>.</li>
</ul>
<pre class="r"><code>qnorm(0.2)</code></pre>
<pre><code>## [1] -0.842</code></pre>
<pre class="r"><code>pnorm(-0.84)</code></pre>
<pre><code>## [1] 0.2</code></pre>
<p><code>r</code>: provide random number generation from a
distribution</p>
</div>
<ol start="2" style="list-style-type: decimal">
<li>Austria had a birth rate of approximately 9.9 births per 1000 people
in 2019. Assume this rate is constant over the year, and that all births
are independent.</li>
</ol>
<ol style="list-style-type: lower-alpha">
<li>For a randomly selected cohort of 1000 people, what is the
probability of observing exactly 11 births over a 1-year period?</li>
</ol>
<div class="soln">
<p>You can model this as a binomial process with <span
class="math inline">\(n=1000,p=0.0099\)</span>, or Poisson, with <span
class="math inline">\(\lambda = 9.9\)</span>.</p>
<pre class="r"><code>c(binom = dbinom(11, 1000, 0.0099),
pois = dpois(11, 9.9))</code></pre>
<pre><code>## binom pois
## 0.113 0.113</code></pre>
</div>
<ol start="2" style="list-style-type: lower-alpha">
<li>What about observing <strong>11 or more</strong> births?</li>
</ol>
<div class="soln">
<p>Lots of ways to arrive at the same answer</p>
<pre class="r"><code># the same logic here applies for binom
c(
## ppois(10, ...) is the prob of 10 or less events
## 1 - ppois(10, ...) gives the prob of more than 10
ppois1 = 1 - ppois(10, 9.9),
## we can achieve the same thing with the lower.tail argument
ppois2 = ppois(10, 9.9, lower.tail=FALSE),
## the poisson is discrete, so we can also sum the mass function
dpois1 = 1 - sum(dpois(0:10, 9.9)),
## technically we need to go to infinity, but the error is small
dpois2 = sum(dpois(11:1e6, 9.9))
)</code></pre>
<pre><code>## ppois1 ppois2 dpois1 dpois2
## 0.404 0.404 0.404 0.404</code></pre>
</div>
<ol start="3" style="list-style-type: decimal">
<li>Is a <strong>probability density</strong> the same as a
<strong>probability</strong>?</li>
</ol>
<ol style="list-style-type: lower-alpha">
<li>Human height is normally distributed within populations. From
1980-1994 within 20 wealthy countries, mean female height was 164.7 cm,
with a standard deviation of 7.1 cm. What is the <strong>maximum
probability density</strong> of this normal distribution, and what is
the x-value <span class="math inline">\(x_{max}\)</span> at which
maximum probability density occurs?</li>
</ol>
<div class="soln">
<p>The maximum density of the normal is at the mean, so <span
class="math inline">\(x_{max}=164.7\)</span></p>
<pre class="r"><code>xmax = 164.7
mean_ht = 164.7
sd_ht = 7.1
dnorm(xmax, mean = mean_ht, sd = sd_ht)</code></pre>
<pre><code>## [1] 0.0562</code></pre>
</div>
<ol start="2" style="list-style-type: lower-alpha">
<li><p>What is the probability that a female in this time period has a
height exactly equal to <span
class="math inline">\(x_{max}\)</span></p></li>
<li><p>If the maximum probability density and the <span
class="math inline">\(pr(x_{max})\)</span> are not the same, why not? Do
these answers make sense?</p></li>
</ol>
<div class="soln">
<p>The probability is zero, because <span
class="math inline">\(x\)</span> is continuous. We computed the density
in part a. We can compute the actual probability (probability mass) by
integrating; clearly it will be zero, because the integral has a width
of zero.</p>
<pre class="r"><code>integrate(dnorm, lower = xmax, upper = xmax, mean = mean_ht, sd = sd_ht)</code></pre>
<pre><code>## 0 with absolute error < 0</code></pre>
</div>
<ol start="4" style="list-style-type: lower-alpha">
<li>What is the probability that a woman has a height in the range <span
class="math inline">\(x_{max} \pm 3\)</span></li>
</ol>
<div class="soln">
<p>There is nonzero mass between two different values. We can either
integrate as before, or use pnorm**</p>
<pre class="r"><code>c(pnorm = pnorm(xmax + 3, mean_ht, sd_ht) - pnorm(xmax - 3, mean_ht, sd_ht),
integrate = integrate(dnorm, lower = xmax - 3, upper = xmax + 3, mean = mean_ht, sd = sd_ht)["value"])</code></pre>
<pre><code>## $pnorm
## [1] 0.327
##
## $integrate.value
## [1] 0.327</code></pre>
</div>
<ol start="4" style="list-style-type: decimal">
<li>For the same distribution, what is the 40th percentile for height?
In other words, what is the value <span class="math inline">\(x\)</span>
such that the probability of observing x or less is 0.4? — <span
class="math inline">\(pr(X \le x) = 0.4\)</span></li>
</ol>
<div class="soln">
<pre class="r"><code>qnorm(0.4, 164.7, 7.1)</code></pre>
<pre><code>## [1] 163</code></pre>
</div>
<ol style="list-style-type: lower-alpha">
<li>What is <span class="math inline">\(x\)</span> if <span
class="math inline">\(pr(X > x) = 0.4\)</span>?</li>
</ol>
<div class="soln">
<pre class="r"><code>qnorm(0.4, 164.7, 7.1, lower.tail=FALSE)</code></pre>
<pre><code>## [1] 166</code></pre>
<pre class="r"><code>qnorm(0.6, 164.7, 7.1, lower.tail=TRUE)</code></pre>
<pre><code>## [1] 166</code></pre>
</div>
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