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.

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