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Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution. - CBSE (Science) Class 12 - Mathematics

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Question

Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution.

Solution

First six positive integers are {1, 2, 3, 4, 5, 6}
No. of ways of selecting 2 numbers from 6 numbers without replacement = 6C2 = 15
X denotes the larger of the two numbers, so X can take the values 2, 3, 4, 5, 6.
Probability distribution of X:

X 2 3 4 5 6
P(x) 1/15 2/15 3/15 4/15 5/15

Computation of Mean and Variance:

Xi P(X=xi) pixi pixi2
2 1/15 2/15 4/15
3 2/15 6/15 18/15
4 3/15 12/15 48/15
5 4/15 20/15 100/15
6 5/15 30/15 180/15
   

`sum p_ix_1=70/15=14/3`

`sump_ix_^2=350/15=70/3`

 

Mean `=sump_ix_i=70/15=4.67`

Variance `=sump_ix_i^2-(sump_ix_i)^2=70/3-196/9=(210-196)/9=14/9`

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Solution Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution. Concept: Mean of a Random Variable.
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