Class 12 Maths Probability | Random Variables and its probability distribution |

**Random Variables and its probability distributions**:

A random variable is a real valued function whose domain is the sample space of a random experiment.

The probability distribution of a random variable X is the system of numbers

where Pi > 0, i=1 to n and P1+P2+P3+ …..... +Pn =1

* Example*: Suppose that a coin is tossed twice so that the sample space is S = {HH,HT,TH,TT}. Let X represent the number of heads that can come up. So with each sample point we can associate a number for X so for HH the value of X is 2 as in HH there are 2 heads, for HT the value of X is 1 as there is only one head in HT. similarly for TH, X=1 and for TT value of X = 0 as there are no heads in TT.

So in the sample space S = {HH,HT,TH,TT} assuming that the coin is fair,

P(HH)= ¼ , P(HT)= ¼ , P(TH)= ¼ , P(TT)= ¼

therefore P(X=0) = P(TT) = ¼

P(X=1) = P(HT ∪ TH) = P(HT) + P(TH) = ¼ + ¼ = ½

P(X=2) = P(HH) = ¼

Thus the table formed is shown below:

* Example*: A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

* Solution*:

Let the probability of getting a tail in the biased coin be x.

∴ P (T) = x

⇒ P (H) = 3x

For a biased coin, P (T) + P (H) = 1

⇒ x+3x=1

⇒4x=1

⇒x=1/4

∴P (T)= 1/4 and P(H)= 3/4

When the coin is tossed twice, the sample space is {HH, TT, HT, TH}.

Let X be the random variable representing the number of tails.

∴ P (X = 0) = P (no tail) = P (H) × P (H)= 3/4 ×3/4

= 9/ 16

P (X = 1) = P (one tail) = P (HT) + P (TH) = 3/4 ×1/4 +1/4 ×3 /4

= 3/16 + 3/16

=3/8

P (X = 2) = P (two tails) = P (TT)= 1/4 ×1/4 = 1/16

Therefore, the required probability distribution is as follows.

.