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A random variable x has following probability distribution:


Values of x:

0

1

2

3

4

5

6

7

8

P(x):

a

3a

5a

7a

9a

11a

13a

15a

17a


Determine the value of k, if $a = \dfrac{1}{k}$ .


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Last updated date: 22nd Mar 2024
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MVSAT 2024
Answer
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Hint: Here we will apply the property of the probability i.e. Sum of all probabilities is equal to one.

Complete step-by-step answer:
In a probability distribution the sum of all probabilities is equal to one.
$ \Rightarrow \sum\limits_{x = 0}^n {P(x)} = 1$
Here, n=8. Therefore,
$
   \Rightarrow \sum\limits_{x = 0}^8 {P(x)} = a + 3a + 5a + 7a + 9a + 11a + 13a + 15a + 17a = 1 \\
   \Rightarrow 81a = 1 \\
   \Rightarrow a = \dfrac{1}{{81}} \\
$
Now it is given that $a = \dfrac{1}{k}$
$ \Rightarrow \dfrac{1}{{81}} = \dfrac{1}{k}$
So on comparing $k = 81$.

Note: In such types of questions the key concept we have to remember is that the sum of the probability distribution is always 1 so, simply add all the probabilities and equate to 1.Probability for a particular value or range of values must be between 0 and 1.