Solved Problems On Random Variables

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Random variables can be any outcomes from some chance process, like how many heads will occur in a series of 20 flips.

We calculate probabilities of random variables and calculate expected value for different types of random variables.

A random process is (just like you would guess) an event or experiment that has a random outcome.

For example: rolling a die, choosing a card, choosing a bingo ball, playing slot machines or any one of hundreds of thousands of other possibilities. Let’s say you wanted to know how many sixes you get if you roll the die a certain number of times.

Your score $X$ on the exam is the total number of correct answers. What is $P(X The number of customers arriving at a grocery store is a Poisson random variable. Let $X$ be the number of customers arriving from $10am$ to $am$.

What is $P(10 Let $X$ be a discrete random variable with the following PMF \begin \nonumber P_X(k) = \left\{ \begin \frac & \quad \text k=-2\ \frac & \quad \text k=-1\ \frac & \quad \text k=0\ \frac & \quad \text k=1\ \frac & \quad \text k=2\ 0 & \quad \text \end \right.If you're seeing this message, it means we're having trouble loading external resources on our website.If you're behind a web filter, please make sure that the domains *.and *.are unblocked.So for example, if you cared about the probability that the sum of the upward faces after rolling seven dice-- if you cared about the probability that that sum is less than or equal to 30, the old way that you would have to have written it is the probability that the sum of-- and you would have to write all of what I just wrote here-- is less than or equal to 30. And then you would try to figure it out somehow if you had some information.But now we can just write the probability that capital Y is less than or equal to 30. And if someone else cares about the probability that this sum of the upward face after rolling seven dice-- if they say, hey, what's the probability that that's even, instead of having to write all that over, they can say, well, what's the probability that Y is even?You solve for the value of x, and x therefore represents a particular number (or set of numbers, if you’re talking about a function).Then you get to statistics and different kinds of variables are used, including random variables.And random variables at first can be a little bit confusing because we will want to think of them as traditional variables that you were first exposed to in algebra class. This is actually a fairly typical way of defining a random variable, especially for a coin flip. It might not be as pure a way of thinking about it as defining 1 as heads and 0 as tails. Notice we have taken this random process, flipping a coin, and we've mapped the outcomes of that random process. And when we talk about the sum, we're talking about the sum of the 7-- let me write this-- the sum of the upward face after rolling 7 dice.And that's not quite what random variables are. So I'm going to define random variable capital X. So random variable capital X, I will define it as-- It is going to be equal to 1 if my fair die rolls heads-- let me write it this way-- if heads. Once again, we are quantifying an outcome for a random process where the random process is rolling these 7 dice and seeing what sides show up on top.But the simple way of thinking about it is as soon as you quantify outcomes, you can start to do a little bit more math on the outcomes.And you can start to use a little bit more mathematical notation on the outcome.

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