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    Anyone know how to do these? I feel like they're a lot more straight forward than I realise...

    Q2) A random variable X has the following distribution:



    a) Calculate its expected value.

    b) Calculate its variance.

    c) Work out the distribution of X2.

    d) Find the expected value of X2.

    e) Find the variance of X2.




    Q3) ​You are catering for 16 people and provide 4 vegetarian and 12 non-vegetarian meals. (Hint: you are assuming that the probability someone is a vegetarian is 4/16 ie 0.25. You also assume that meat eaters do not want a vegetarian meal). You can answer this with Excel using the function ‘Statistical’ and then ‘BINOMDIST’ option. Or, you can answer it using statistical tables.

    a)​What is the probability that you will have at least one disappointed guest?

    b)​What is the probability that there is at least one disappointed vegetarian guest but no disappointed non-vegetarians?

    c)​Suppose you were to provide 17 meals for 16 guests. Would it be more sensible to provide an extra vegetarian meal or non-vegetarian?

    d)​If you provide an extra non-vegetarian meal (i.e 17 meals for 16 people), what is the probability that at least one person will be disappointed?


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    (Original post by JellyCat99)
    Anyone know how to do these? I feel like they're a lot more straight forward than I realise...

    Q2) A random variable X has the following distribution:
    ?

    In either case, if a R.V has a pdf: f(x), then the expected value is given by \displaystyle \int_{-\infty}^{\infty} xf(x) \, \mathrm{d}x
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    Just to give you some more background on Zack's answer, the  f(x) \, \mathrm{d}x part is the probability of the random variable taking an infinitesimal region in the sample space.

    As an example, just imagine looking at a window when its raining and seeing all these different sizes. You then ask the question, "what is the probability of the rain drop being between 1mm and 2mm". Now, to get that probability, we would have:

    \displaystyle \int_{1}^{2} xf(x) \, \mathrm{d}x. In this case, x would be represent the sample space of ALL the possible droplet sizes. We then restrict the sample size to be between 1mm and 2mm and thus we get our probability.If we have limits of negative to positive infinity, then we are simply looking at the entire sample space. So back to our example, rather than looking at the probability of drop sizes between 1mm and 2mm, we are looking at the probability of getting droplet sizes between negative infinity and positive infinity. This has to be one.

    When we perform integration, one interpretation of it is finding the area under the graph. So when we perform the integration between negative infinity and positive infinity, we find the area to be one. Going to our example, this means all the probabilities of all the rain droplet sizes are equal to one. This makes sense because we are certain our droplet sizes will be within this range of sizes and a "certainty" in probability is one.

    Moving onto the variance, we use a similar formula, except this time, we have to do some operations on the random variable before we multiply it with its probability,  f(x) \, \mathrm{d}x.

    The key take home point is that  f(x) is called the Probability Density Function (PDF). It may be easier to first understand the Cumulative Distribution Function (CDF). The CDF is a function with its input, a random variable assigned a specific value. So lets say our random variable is a measure of droplet sizes and we've assigned it a droplet size of 1.5mm. If we give that as an input into the CDF, it would tell us the probability of all the droplet sizes being less then or equal to 1.5mm. This is powerful because if we assign instead, the random variable to be equivalent to infinity, you'll see the output from the CDF must be one, i.e. the probability is certain. The PDF is then simple the derivative of the CDF.

    I like to think of the PDF (though this is probably incorrect) as the measure of how the probability of an event on the sample space, x, is changing along the sample space,
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    (Original post by djpailo)
    Just to give you some more background on Zack's answer, the  f(x) \, \mathrm{d}x part is the probability of the random variable taking an infinitesimal region in the sample space.
    Nice explanation, +rep.
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    I cannot see the distribution of X in Q1
    is it discrete or continuous ?
 
 
 
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