A Random Variable is a set of possible values from a random experiment. Show
Let's give them the values Heads=0 and Tails=1 and we have a Random Variable "X": In short: X = {0, 1} Note: We could choose Heads=100 and Tails=150 or other values if we want! It is our choice. So:
Not Like an Algebra VariableIn Algebra a variable, like x, is an unknown value:
In this case we can find that x=4 But a Random Variable is different ... A Random Variable has a whole set of values ...... and it could take on any of those values, randomly.
X could be 0, 1, 2, or 3 randomly. And they might each have a different probability. Capital LettersWe use a capital letter, like X or Y, to avoid confusion with the Algebra type of variable. Sample SpaceA Random Variable's set of values is the Sample Space.
Example: Throw a die onceRandom Variable X = "The score shown on the top face". X could be 1, 2, 3, 4, 5 or 6 So the Sample Space is {1, 2, 3, 4, 5, 6} ProbabilityWe can show the probability of any one value using this style: P(X = value) = probability of that value
X = {1, 2, 3, 4, 5, 6} In this case they are all equally likely, so the probability of any one is 1/6
Note that the sum of the probabilities = 1, as it should be.
X = "The number of Heads" is the Random Variable. In this case, there could be 0 Heads (if all the coins land Tails up), 1 Head, 2 Heads or 3 Heads. So the Sample Space = {0, 1, 2, 3} But this time the outcomes are NOT all equally likely. The three coins can land in eight possible ways:
Looking at the table we see just 1 case of Three Heads, but 3 cases of Two Heads, 3 cases of One Head, and 1 case of Zero Heads. So:
Example: Two dice are tossed.The Random Variable is X = "The sum of the scores on the two dice". Let's make a table of all possible values:
There are 6 × 6 = 36 possible outcomes, and the Sample Space (which is the sum of the scores on the two dice) is {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Let's count how often each value occurs, and work out the probabilities:
A Range of ValuesWe could also calculate the probability that a Random Variable takes on a range of values.
In other words: What is P(5 ≤ X ≤ 8)?
P(5 ≤ X ≤ 8) =P(X=5) + P(X=6) + P(X=7) + P(X=8) = (4+5+6+5)/36 = 20/36 = 5/9 SolvingWe can also solve a Random Variable equation.
Looking through the list above we find:
So there are two solutions: x = 4 or x = 10 Notice the different uses of X and x:
ContinuousRandom Variables can be either Discrete or Continuous:
All our examples have been Discrete. Learn more at Continuous Random Variables. Mean, Variance, Standard DeviationYou can also learn how to find the Mean, Variance and Standard Deviation of Random Variables. Summary
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