Question

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 $a = \dfrac{1}{k}$ .

Answer 1

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.