Worksheet oblems 6 Expected Value therandomvariableisandmeans givenyoumustshoworcalculatetheprobabilities ectedvalueequationtheprobabilitiesmustadduptoone retingyouranswer.(practicalterms)
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Notes for Random Variable and Expected Value: READ slides 52-63, focus on the project 67-69 Fill in the chart below before class Example 5: Toss a fair two-sided coin 3 times or 3 coins once [Trial] Let X record the number of heads on the 3 tosses. X is a random variable (denoted with a capital letter). Toss 3 coins and count the total number of heads. (Let X be the total number of heads in 3 tosses of a coin.) Repeat 20 times [20 trials] and record the total number of heads in each of the 20 trial of this experiment. (You are simulating 20 values of X).
Total number of heads=_______ Random variable is a rule that assigns a numerical value to each outcome of an experiment. From TEXT: Given an experiment, a random variable, X, and a number, x, the expression X = x stands for the event that an outcome occurs to which X is assigned the value x. Example 1 Suppose we toss a fair coin repeatedly until a Head occurs and then we stop. Let the random variable be Y denote the number of tosses preformed in the experiment. Sample Space is The value of Y The numbers 1, 2, 3, are y
means
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Worksheet 6 Expected Value
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Worksheet 6 Expected Value 3 Example 2 Suppose we toss a fair coin three times and introduce the random variable X to denote the number of heads that occur in the three tosses. Sample space is The value of X The random variable X takes on the values of x = {0, 1, 2, 3} Consider the sample space SH ={0, 1, 2, 3} The first example Y is an infinite discrete random variable. The second example X is a finite discrete random variable Example 2 P(X =1), means
P(X 1), means From the text:
Suppose that a random variable X can assume only the distinct values x1, x2, , xn. At least one of the events X = x1, X = x2, , X = xn must occur on any trial of the experiment and no two of the events can both occur at the same time. Thus, 1 = P(X = x1 X = x2 X = xn) and
Example 3a Tossing a fair coin 3 times (or 3 coins once) and recording the number of heads Sample space:
We have 8 possible outcomes for this experiment. If we assume the coin is fair, then each of these 8 outcomes will be equally likely. P(HHH)=P(TTT)=P(HTH)=P(TTH) etc Because of independence P(HHH)=P(H)P(H)P(H)=(1/2)(1/2)(1/2)=1/8=.125
Worksheet 6 Expected Value X = the number of Heads one gets when a coin is tossed 3 times Making a table: (You will use this many times) xi 0 1 2 3 P(X = 1) The sum of all the probabilities is 1 you must always verify this It is important to keep upper case X and lower case x different in your mind. You must declare what the random variable is and what it means. Example 3b Tossing a bias coin 3 times (or 3 coins once) and recording the number of heads Sample set of outcomes: HHH HHT TTH TTT HTH THT THH HTT The probabilities of these are different because not equally likely P(HHH)P(HHT) etc We still have 8 possible outcomes for this experiment. If the bias coin has a 40% of getting a Head Means P(H)=.4 and P(T) = .6 X = the number of Heads one gets when a coin is tossed 3 times Making a table: (You will use this many times) xi 0 1 2 3 P(X =xi) P(X =0)= 1(.6)(.6)(.6)=0.216 P(X =1)= 3(.4)(.6)(.6)= 0.432 P(X =2)= 3(.4)(.4)(.6)= 0.288 P(X =3)= 1(.4)(.4)(.4)=0.064 P(X =xi) P(X =0)= P(X =1)= P(X =2)= P(X =3)=
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The sum of all the probabilities is 1 P(X1)=
Worksheet 6 Expected Value
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Expected Value: E(X)
READ 52-63, slides focus on the project 67-69 Just as an introduction Example 4 What if I said we are going to play a game. A fair coin will be flipped. Heads you pay me $5 and Tails you will win $2. Do you want to play? How many times? Do you think this is a fair game? Why? The probability of heads and tails are the same. So why is it not a fair game? From the text The expected value of X, denoted by E(X), is the sum of these values, weighted by their respective probabilities. The expected value E(X) of a finite discrete random variable is the ____________________ of X when the experiment is repeated many times. Suppose a random variable X can only assume the distinct values x1, x2, , xn. We can compute the average (mean) value of X. The mean value for the random variable is given by
We call this mean value the expected value of X. It is denoted by either E(X) or X . Example 5: Toss a fair two-sided coin 3 times or 3 coins once [Trial] Let X record the number of heads on the 3 tosses. X is a random variable (denoted with a capital letter). Find the expected value. From page 1 record the number of heads you got Total number of heads=_______ Average number of heads in 3 tosses= Total # of heads/20= estimate of E(X)=_____ Lets calculate the actual expected value of flipping a fair coin 3 times and recording the number of Heads: Example 3a xi P(X =xi) 0 P(X =0)= 1/8 1 P(X =1)= 3/8 2 P(X =2)= 3/8 3 P(X =3)= 1/8 Expected value:
E(X)
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Worksheet 6 Expected Value
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Example 5b: Using example 3b calculate the expected value of the bias coin P(H)=.4
Finishing Example 4 What if I said we are going to play a game. A fair coin will be flipped. Heads you pay me $5 and Tails you will win $2. Do you want to play?
Note: This game is the same if it was written. A player pays $5 per game. A coin is flipped. Heads you lose(gain nothing), Tails the player wins $7. (Everyone lose $5 upfront)
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Example 6 Ms. Schuster is the host of a new television game show called Do you want to Play? Heres how you play the game: The player tosses a fair 6 sided die one time. If the die lands with 1 dot on the face, you lose and have...