The number of children per household, x, in the United...

The number of children per household, $x,$ in the United States in 2008 is expressed as a probability distribution here. $$\begin{array}{l|cccccc}\hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5+ \\\boldsymbol{P}(\boldsymbol{x}) & 0.209 & 0.384 & 0.249 & 0.106 & 0.032 & 0.020 \\\hline\end{array}$$ a. Is this a discrete probability distribution? Explain. b. $\quad$ Draw a histogram for the distribution of $x,$ the number of children per household. c. Replacing "5+" with exactly "5," find the mean and standard deviation.

The number of children per household, $x,$ in the United States in 2008 is expressed as a probability distribution here.
$$\begin{array}{l|cccccc}\hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5+ \\\boldsymbol{P}(\boldsymbol{x}) & 0.209 & 0.384 & 0.249 & 0.106 & 0.032 & 0.020 \\\hline\end{array}$$
a. Is this a discrete probability distribution? Explain.
b. $\quad$ Draw a histogram for the distribution of $x,$ the number of children per household.
c. Replacing "5+" with exactly "5," find the mean and standard deviation.

Key Concepts

Standard Deviation Calculation

The standard deviation is a measure of the dispersion or spread of the values around the mean in a probability distribution. For a discrete random variable, it is computed by finding the square root of the variance, which itself is calculated by taking the weighted average of the squared differences between the outcomes and the mean. This metric quantitatively expresses how much the outcomes typically deviate from the average value.

Expected Value (Mean) Calculation

The expected value or mean of a discrete random variable is calculated as the sum of the products of each outcome and its corresponding probability. It provides a measure of the central tendency of the distribution. In the given problem, after adjusting the category '5+' to an exact value of 5, the expected value is obtained by multiplying each number of children by its probability and summing these products.

Discrete Probability Distribution

A discrete probability distribution is one in which the random variable takes on a countable number of distinct values, each with an associated probability. In this context, x represents the number of children per household, and each possible outcome (0, 1, 2, 3, 4, and 5 or more) is assigned a specific probability. One key requirement is that the sum of these probabilities equals 1, confirming it as a valid probability distribution.

Histogram

A histogram is a graphical representation of the distribution of a dataset where the x-axis displays the different outcomes or categories and the y-axis represents the frequency or probability of each outcome. For a discrete probability distribution, each bar corresponds to one specific outcome and its height is proportional to the probability of that outcome. This visualization helps in understanding the distribution's shape and spread.

Transcript

00:01 In this question, we are given the following probability distribution that i have written out.

00:06 And part a asks us, is this a discrete probability distribution? and the answer is no because of the 5 plus.

00:18 So here we have 5 plus.

00:21 And that 5 plus could be anything from 5 and a half to hundreds to thousands, anything.

00:27 Obviously thousands of children isn't that realistic.

00:30 But anything above five is counts in that, which is why it's not discrete.

00:35 And that is the answer for that one.

00:38 So be asked us to draw a histogram for this data.

00:43 So i'm going to draw that here because i'll just draw it.

00:55 I'll just draw it underneath.

00:56 So we've got one, zero, one, two, three, four and five plus.

01:07 So we've got 0, 1, 2, 3, 4, 5 plus.

01:13 So to do a histogram, we then need to put in our percent, which are our probabilities times 100.

01:21 And we can see the biggest one is 24.

01:24 So if you do 5, 10, 15, 20, 25.

01:29 Because remember, you don't always need to go all the way to 100.

01:32 Because in this case, that would be a bit of a waste of space, because nothing goes above 25.

01:40 So 0 .209, which is just over 20%.

01:45 So about there.

01:47 And when you do yours, probably do it on graph paper or with a ruler, so it's slightly more accurate than mine.

01:56 Sorry, it was my mistake here.

01:57 It actually goes up to 38.

02:00 So 25, 30, 35.

02:04 So the next one is 38, which if that's 30, that's 35, 3538 is the very top of our graph.

02:15 The next one is 24 .9, which will be about just under 25.

02:26 3 is 10 .6, just above 10.

02:32 4 is 0 .032, so that's only just 3 .2%.

02:38 And then a 5 is plus is even smaller.

02:42 So that is the shape for our graph.

02:44 And if we did a little shape, we'd get something that looks a little bit like that.

02:48 If you get asked to describe the shape, you can say it has a skew towards zero and one, and stuff like that.

02:57 So that is part b.

02:59 So now we're replacing five with exactly five, that's five plus.

03:06 So we're going to scrub that bit and just say this is five, otherwise this next bit's impossible...

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