Can someone please explain to me in layman's terms what a joint distribution is? I do not understand it after seeing a word problem that pertained to joint distributions. Please provide the intuition and avoid mathematical equations.

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    $\begingroup$ For a discrete distribution of a random variable $X$, you can consider $\Pr(X=x)$ for different values of $x$. For a discrete joint distribution of random variables $X$ and $Y$, you can consider $\Pr(X=x \text { and }Y=y)$ for different values of $x$ and $y$. You need something more sophisticated for random variables with continuous distributions, but the intuitive idea is similar $\endgroup$ – Henry May 7 '17 at 11:05

As a concrete example, suppose I toss a coin and roll a die one after the other. As you know, there is a probability distribution associated with the outcomes of both (discrete uniform distributions, i.e. every possible outcome is equally likely). I can use these to find e.g. P(coin lands heads) or P(I roll a six).

The joint probability distribution is just a distribution over combinations of these events. So it tells me about P(I roll a six and the coin lands heads), or P(the coin lands tails and I roll a two).

As a continuous example, IQs are distributed normally with mean 100 and adult male heights are distributed normally with mean about $5' 10''$. The joint distribution is then a distribution over (height, IQ) pairs, so it tells you about the probability of having both a given height and a given IQ, rather than just one or the other. So armed with the joint distribution, I can now ask questions like 'what is the probability that a man off the street has an IQ between 90 and 110 and is between 5'9'' and 5'11' tall.'

In short, you make a new event, which is the combination of two events you already know about. The joint distribution tells you about the probability of the new event.

Edit: As noted in the comments, when you are computing joint distributions, it's very important to think about whether the individual components are independent. Say we want to find the joint distribution over house prices and number of bedrooms in your neighbourhood. The result will still be a distribution over (house price, number of bedroom) pairs, just as above. However, you will need to explicitly account for the fact that these are not independent by using a conditional probability. This is what makes joint distributions really useful -- they account for the influence of one component on the other. Note also that in this example, the joint density is mixed, i.e. one component is continuous (house price) and one is discrete (number of bedrooms). We can still use it to ask, 'What is the probability that a house has 4 bedrooms and is worth between £400,000 and £500,000?'

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    $\begingroup$ +1, but it may have helped to use an example where the two quantities aren't (basically) independent! $\endgroup$ – Danica May 7 '17 at 11:58
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    $\begingroup$ Good point, I've updated :) $\endgroup$ – Will May 7 '17 at 12:25

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