Discrete Random Variables: Probability & Statistics

Learning Objectives 1. Understand the concepts of a random variable and a probability distribution. 2. Be able to distinguish between discrete and continuous random variables. 3. Calculate probabilities for multiple random variables based on joint probability functions 4. Be able to compute and interpret the expected value, variance, and standard deviation for a discrete random variable.
DISCRETE RANDOM VARIABLES

DISCRETE RANDOM VARIABLES
DEFINITION :Adiscrete random variable is a function X(s)from
afinite or countably infinite sample space Sto the real numbers :
X(·):S→R.
EXAMPLE : Toss a coin 3 times in sequence. The sample space is S={HHH , HHT , HTH , HTT , THH , THT , TTH , TTT}, and examples of random variables are
•X(s) = the number of Heads in the sequence ; e.g.,X(HTH)=2,
•Y(s) = The index of the first H;e.g.,Y(TTH)=3,
0 if the sequence has no H,i.e.,Y(TTT)=0.
NOTE :InthisexampleX(s)andY(s) are actually integers .
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Value-ranges of a random variable correspond to events in S. EXAMPLE : For the sample space S={HHH , HHT , HTH , HTT , THH , THT , TTH , TTT}, with
X(s) = the number of Heads ,
the value
X(s)=2,corresponds to the event {HHT , HTH , THH},
and the values 1<X(s)≤3,correspond to {HHH , HHT , HTH , THH}.
NOTATION : If it is clear what Sis then we often just write
Xinstead of X(s).
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Value-ranges of a random variable correspond to events in S,
and events in Shave a probability . Thus
Value-ranges of a random variable have a probability .
EXAMPLE : For the sample space S={HHH , HHT , HTH , HTT , THH , THT , TTH , TTT}, with X(s) = the number of Heads , we have
P(0 <X≤2) = 6
8.
QUESTION : What are the values of P(X≤−1) ,P(X≤0) ,P(X≤1) ,P(X≤2) ,P(X≤3) ,P(X≤4) ?
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NOTATION : We will also write pX(x)todenoteP(X=x).
EXAMPLE : For the sample space S={HHH , HHT , HTH , HTT , THH , THT , TTH , TTT}, with X(s) = the number of Heads , we have
pX(0) ≡P({TTT})=
1 8
pX(1) ≡P({HTT , THT , TTH})=
3 8
pX(2) ≡P({HHT , HTH , THH})=3
8
pX(3) ≡P({HHH})=
1
8
where
pX(0) + pX(1) + pX(2) + pX(3) = 1 .(Why ? )
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