Introduction to binomial random variable:
Binomial Random variables help us to make a link between probability and numbers that we observe as data.
Binomial Random Variable: A numerical valued function defined on a sample space. A random variable X “maps” an outcome in a sample space to a numerical value. We use capital letters such as X or Y to denote binomial random variables. Let s be an elementary outcome. A value, X(s), of X is denoted x.
A binomial random variable is discrete if it can take on a finite or countable number of values. A continuous random variable takes on an uncountable number of values.
Binomial Random Variable:
Ex 1: What is the probability we roll less than a 5?
Sol : P(X < 5) = P(X = 2) + P(X = 3) + P(X = 4)
= 1/36 + 2/36 + 3/36
= 6/36
the probability we roll less than a 5 = 1/6
Ex 2 : What is the probability we roll a number between 7 and 10 (inclusive)?
Sol : P(7 `<=` X `<=` 10) = P(X = 7) + P(X = 8) + .......
= 6/36 + 5/36 + 4/36 + 3/36
= 18/36
the probability we roll a number between 7 and 10 (inclusive) = 1/2
Ex 3 : What is the probability the sum of two dice will be odd?
Sol : P(X odd) = P(X = 3) + P(X = 5) + ..........
= 2/36 + 4/36 + 6/36 + 4/36 + 2/36
= 18/36
the probability the sum of two dice will be odd = 1/2
Ex 4: What is the probability we roll a number between 8 and 12 ?
Sol : P(8 < X <12 10="10" 11="11" 9="9" br="br" p="p">
= 8/36 + 9/36 + 10/36
= 27/36
the probability we roll a number between 8 and 12 = 3/4
Binomial Random variables help us to make a link between probability and numbers that we observe as data.
Binomial Random Variable: A numerical valued function defined on a sample space. A random variable X “maps” an outcome in a sample space to a numerical value. We use capital letters such as X or Y to denote binomial random variables. Let s be an elementary outcome. A value, X(s), of X is denoted x.
A binomial random variable is discrete if it can take on a finite or countable number of values. A continuous random variable takes on an uncountable number of values.
Binomial Random Variable:
General features of a Binomial Random Variable:
The binomial random experiment consists of n identical trials.
Possible outcomes on each trial are of only two. given one outcome by S (for Success) and the other by F (for Failure).
The probability for Success(S) remains the same from trial to trial. Denote it by p=Pr(Success).
The trials of binomial random experiment are independent of each other.
Then we say Binomial Random Variable X as the number of Successes in n trials.
The binomial random experiment consists of n identical trials.
Possible outcomes on each trial are of only two. given one outcome by S (for Success) and the other by F (for Failure).
The probability for Success(S) remains the same from trial to trial. Denote it by p=Pr(Success).
The trials of binomial random experiment are independent of each other.
Then we say Binomial Random Variable X as the number of Successes in n trials.
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Binomial Random Variable:
Ex 1: What is the probability we roll less than a 5?
Sol : P(X < 5) = P(X = 2) + P(X = 3) + P(X = 4)
= 1/36 + 2/36 + 3/36
= 6/36
the probability we roll less than a 5 = 1/6
Ex 2 : What is the probability we roll a number between 7 and 10 (inclusive)?
Sol : P(7 `<=` X `<=` 10) = P(X = 7) + P(X = 8) + .......
= 6/36 + 5/36 + 4/36 + 3/36
= 18/36
the probability we roll a number between 7 and 10 (inclusive) = 1/2
Ex 3 : What is the probability the sum of two dice will be odd?
Sol : P(X odd) = P(X = 3) + P(X = 5) + ..........
= 2/36 + 4/36 + 6/36 + 4/36 + 2/36
= 18/36
the probability the sum of two dice will be odd = 1/2
Ex 4: What is the probability we roll a number between 8 and 12 ?
Sol : P(8 < X <12 10="10" 11="11" 9="9" br="br" p="p">
= 8/36 + 9/36 + 10/36
= 27/36
the probability we roll a number between 8 and 12 = 3/4
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