A random variable is a function which is usually denoted by X defined on the sample space S whose range is the set of real numbers.
i.e. X:S–> R is called a random variable.
Example:
If an unbiased coin is tossed,
Then sample space S={H, T}
Let X denoted the number of tails obtained.
Then X(T)=1 (the number of chances of getting ‘T’ =1).
Probability distribution:
The probability distribution of a random variable x is as follows.
|
X = r |
1 |
2 |
3 |
……. |
n |
|
P(X=r) |
P(X=1) |
P(x=2) |
P(X=3) |
………….. |
P(X=n) |
Here all probabilities are non – zero and sum of all probabilities is 1.
Example – 1:
If 2 coins are tossed and X is the random variable which denotes the number of heads then
Sample space, S = {HH,HT,TH,TT}
n(S) = 4.
X can be 0,1,2
P(X=0) = probability of getting zero heads = 1/4 (TT)
P(X=1) = probability of getting 1 head = 2/4=1/2 (HT,TH)
P(X=2) = probability of getting 2 heads = 1/4 (HH)
So, the probability distribution of X is as follows.
|
X = r |
0 |
1 |
2 |
|
P(X=r) |
1/4 |
1/2 |
1/4 |
Example-2:
If 3 coins are tossed and X is the random variable which denotes the number of heads then
Sample space, S = {HHH,HHT,HTT,TTT,TTH,THH,HTH,THT}
n(S) = 8.
X can be o,1,2,3
P(X=0) = probability of getting zero heads = 1/8 (TTT)
P(X=1) = probability of getting 1 head = 3/8 (HTT,THT,TTH)
P(X=2) = probability of getting 2 heads = 3/8 (HHT,HTH,THH)
P(X=3) = probability of getting 3 heads = 1/8 (HHH)
So, the probability distribution of X is as follows.
|
X = r |
0 |
1 |
2 |
3 |
|
P(X=r) |
1/8 |
3/8 |
3/8 |
1/8 |
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