How do you validate a pmf? [closed]

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$$P\left(X=x\right)=\left\{ \begin{array}{rl} 0.02 & X=0\\ 0.03 & X=1\\ 0.2 & X\in\left\{ 2,\,5\right\} \\ 0.25 & X\in\left\{ 3,\,4\right\} \\ c & X=6 \end{array}\right.$$

I have the above distribution and from it I am to answer the following questions

a Show that this is a valid pmf (you will need to calculate c for this).

b What is the probability that the chef will require at most 4 chickens?

c Find the expected number of chickens that the chef will need to buy.

I am unsure how to answer question a as I have not done one where the is a variable.

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1 Answer

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Any valid probability mass function is required to have certain properties.

So if $\mathsf P(X{=}x)$ is a probability mass function with support $\{0,1,2,3,4,5,6\}$, then:

• For any discrete value in the support of the function, the image of the function lies in the interval $[0;1]$. So:$$\forall x\in\{0,1,2,3,4,5,6\}~\Big(0\leqslant \mathsf P(X{=}x)\leqslant 1\Big)$$
• The sum of probabilities for every value in the support equals one.$$\sum_{x=0}^6\mathsf P(X{=}x) = 1$$Thus you are required to determine the value for $c$ which makes these things true.

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