Geometric Random Variable
Introduction to geometric random variable :
The branch of mathematics which deals with the measurements of solids, lines, angles, surfaces etc.For example the sequence 2, 6, 18, 54, is a geometric progression with common ratio 3. Geometry means not only study of the angles and triangles, perimeter, area and volume but also their submission in a mixture of fields. Let us learn about solving geometric random variables.
Problems in Geometric Random Variable:
Problem in geometric random variable1: The probability that a basketball player makes a free throw is 0.5. Find the probability that the player makes of his first free throw on his 5th attempt.
Sol:
For this example p = 0.5 and x = 5. The probability that the first success occur on the fourth attempt is given by
P (x) = (1 p) x-1. p
P (5) = (0.5)3. 0.5
= 0.0625
The probability that the player makes of his first free throw on his fourth attempt is 0.0625
Problem in geometric random variable 2:
When rolling a pair of fair six sided dice, the probability that the sum is 9 or 10 is 0.35, because 10 of the 36 possible rolls have a sum that is 9 or 10. We will construct a geometric probability distribution for the roll containing the first 9 or 10.
Sol:
Success: A roll is an 9 or 10.
Probability of Success: p = 0.35
Failure: A roll is not an 9 or 10.
Probability of failure: 1 p = 0.65
The appropriate formula here is
P(x) = (1 p)x-1 . p
= (0.65)x-1. 0.35
0more Examples in Geometric Random Variable 3:
Problem:
Find the expectation of the number of tosses of the coin, the variable would represent the number of tosses of the coin.
Sol:
Let "x" indicate the number of tosses of the coin
[Since you are required to find the expectation of the number of tosses of the coin, the variable would represent the number of tosses of the coin.]
The number of tosses of the coin would be
1 if a head appears on the 1st throw
2 if a tail appears on the 1st throw and a head appears on the 2nd throw
3 if a tail appears on the 1st 2 throws and a head appears on the 3rd throw
4 if a tail appears on the 1st 3 throws and a head appears on the 4th throw
5 if a tail appears on the 1st 4 throws and a head appears on the 5th throw (Or) if a a tail appears on the 1st 5 throws
"X" is a discrete random variable with range = {1, 2, 3, 4, 5}
"X" represents that the random variable and P(X = x) represents the probability that the value within the range of the random variable is a specified value of "x"
In a single throw with a coin, Probability of:
Getting a head in the first throw = 1/3
Getting a head in the second throw = 2/3 * 1/3 = 2/9
Getting a head in the third throw only = 2/3 * 2/3 *1/3 = 4/27
Getting a head in the fourth throw only = 2/3 * 2/3 * 2/3 * 1/3 = 8/81
Getting a head in the fifth throw only = 2/3 * 2/3 * 2/3 * 2/3 * 1/3 = 16/243
Getting all tails in 5 throws = (2/3)^5 = 32/ 243
The probability distribution of "x" would be
Expected number of toss of coins = 'sum' xp(x) = 1(1/3) + 2 (2/9) + 3(4/27) + 4(8/81) + 5(16/243) = 211/81
= 2.605
Expected number of toss of coins = 2.605 or say 3,
If appearing of head is considered as a success, then
Expected value of the geometric random variable = 1/p = 1/ 1/3 = 3
The above problems are the geometric random variable.
by: Omkar Nayak
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