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Question Video: Finding the Value of an Unknown for a Discrete Random Variable Mathematics

Let π‘₯ denote a discrete random variable that can take the values 0, 1, 2, and 3. Given that 𝑃(π‘₯ = 0) = 1/9, 𝑃(π‘₯ = 1) = 4/9, 𝑃(π‘₯ = 2) = π‘Ž, and 𝑃(π‘₯ = 3) = 3π‘Ž, find the value of π‘Ž.

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Video Transcript

Let π‘₯ denote a discrete random variable that can take the values zero, one, two, and three. Given that the probability π‘₯ equals zero is equal to one-ninth, the probability π‘₯ is equal to one is equal to four-ninths, the probability π‘₯ is equal to two is equal to π‘Ž, and the probability π‘₯ is equal to three equals three π‘Ž, find the value of π‘Ž.

In order to answer this question, we will recall some of the key properties of discrete random variables. A discrete random variable has a countable number of possible values. The probability of each outcome can be described using a probability distribution function. If 𝑓 of π‘₯ tells us the probability of each event occurring, then the sum of all 𝑓 of π‘₯ values must be equal to one. And Each individual 𝑓 of π‘₯ value must be between zero and one inclusive. And in fact, we can use this first property to answer the problem.

We’re told that the probability π‘₯ is equal to zero is equal to one-ninth, the probability π‘₯ is equal to one is four-ninths, the probability π‘₯ is equal to two is π‘Ž, and the probability π‘₯ equals three is equal to three π‘Ž. And we’re also told that these are the only possible outcomes. So the probability that π‘₯ equals zero, one, two, and three, the sum of these is equal to one. We can replace each of these expressions with its corresponding probability value. So one-ninth plus four-ninths plus π‘Ž plus three π‘Ž equals one. One-ninths plus four-ninths is five-ninths. And π‘Ž plus three π‘Ž is four π‘Ž.

We then solve this equation for π‘Ž by subtracting five-ninths from both sides, giving us four π‘Ž equals four-ninths. And finally, we’re going to divide through by four. Well, it follows that four-ninths divided by four is equal to one-ninths. And so given the information about our discrete random variable π‘₯, π‘Ž must be equal to one-ninth.

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