Random Variable

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Random VariablesA random variable, usually written X, is a variable whose possible values

are numerical outcomes of a random phenomenon

exampleFlip a coin three times; X = the total number of heads.The values of X are . X= 0, 1, 2, 3

X= 2, 3, 4, …, 12

X=1, 2, 3, 4, ...

Throw two dice; X = the sum of the numbers facing up.The values of X are .

Throw one die over and over until you get a six; X = the number of throws.The values of X are .

Types of Random Variables

Discrete random variables are ones that have a finite or countable number of possible outcomes (like number of heads when flipping several coins).

Continuous random variables are ones that have an infinite or uncountable number of possible outcomes (like your exact speed on the highway, or how far someone jumps)

Probability with discrete variables

Throw a pair of fair dice, and take X to be the sum of the numbers facing up.

◦ The event that X = 2 is {(1, 1)}     The event that you throw a 2

◦ The event that X = 3 is {(2, 1), (1, 2)}   The event that you throw a 3

◦ The event that X = 4 is {(3, 1), (2, 2), (1, 3)}   The event that you throw a 4

◦ P(X = 4) = 1/12 ◦ The probability that X = 4 is 1/12

X= 2, 3, 4, … ,12

practice

P(X=1) =

P(X=2) =

P(X=3) =

P(X=4) =

P(X=5) =

0

1/36

2/36

3/36

4/36

Problem 2: Tossing a coin 4 times. Find the discrete probability of the following:

let X= number of heads

P(X=0) =

P(X=1) =

P(X=2) =

P(X=3) =

P(X=4) =

1/16

4/16

6/16

4/16

1/16

Throw a pair of fair dice, and take X to be the sum of the numbers facing up.

X= 2, 3, 4, … ,12

Problem 2: Tossing a coin 4 times. Find the probability of the following:

Where: X= number of heads

Number of heads

0 1 2 3 4

Probability 1/16 4/16 6/16 4/16 1/16

P(X ≤ 2)

What is:

= P(x=0) + P(x=1) + P(x=2)

= 1/16 + 4/16 + 6/16

= 11/16 Or 0.6875

Education Level:A study of education followed a large group of

fifth-grade children to see how many years of school they eventually completed. Let X be the highest year of school that a randomly chosen

5th graders completes Years 4 5 6 7 8 9 10 11 12

probability 0.010 0.007 0.007 0.013 0.032 0.068 0.070 0.041 0.752

A. Is this a “legit” continuous probability distribution?

B. What is the percent of 5th graders eventually finished 12th grade?

C. Find P(X≥6)C. Find P(X>6)

Yes, because the probabilities add up to 1

There are 75.2% 5th graders eventually finished 12th grade.

0.983

0.931