Question

If $X$ is an integer-valued random variable, show that the frequency function is related to the cdf by $p(k)=F(k)-F(k-1).$

          If $X$ is an integer-valued random variable, show that the frequency function is related to the cdf by $p(k)=F(k)-F(k-1).$
        

Added by Brittany S.

Elementary Statistics a Step by Step Approach
Elementary Statistics a Step by Step Approach
Allan G. Bluman 9th Edition
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If $X$ is an integer-valued random variable, show that the frequency function is related to the cdf by $p(k)=F(k)-F(k-1).$
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Transcript

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00:01 So for this problem, i'm going to begin by noting that there is a little bit of uncertainty in the notation here.
00:07 I'm gathering by context that p of k is what is being referred to as the frequency function, which may also be referred to as the probability mass function, which would be equivalently written as p of k is equal to the probability.
00:40 I'll use pr to distinguish my indication of probability, it's the probability that our discrete random variable x is equal to a value k.
00:52 Similarly, we have capital f of k.
00:56 I'm assuming that that is the cumulative distribution function, reference, the cdf, where we have that f of k is equal to the probability that x is less than or equal to k.
01:12 So, that being said, we can see that if we take f of k, less than or equal to, or, pardon me, if we take, i need to be careful here with how i say this, if we take f of k minus f of k minus 1, we get that the result should be equal to the probability that x is less than or equal to k minus the probability that x is less than or equal to k minus 1.
01:48 Now when we express probabilities like this, we can actually put those two probabilities together...
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