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A discrete ra...

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A discrete random variable X has the following probability distribution:
X: 1 2 3 4 5 6 7
P(X): c 2c 2c 3c c² 2c² 7c²+c
Then, P(X ≤ 2) = ____________.

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Solution

For a discrete random variable X

X P(X)

1 c

2 2c

3 2c

4 3c

5 c²

6 2c²

7 7c²+ c

$Since \sum_{i = 1}^{7} p (x_{i}) = 1 i . e . c + 2 c + 2 c + 3 c + c^{2} + 2 c^{2} + 7 c^{2} + c = 1 i . e . 8 c + c + 10 c^{2} = 1 i . e . 10 c^{2} + 9 c - 1 = 0 i . e . 10 c^{2} + 10 c - c - 1 = 0 i . e . 10 c (c + 1) - 1 (c + 1) = 0 i . e . c = \frac{1}{10}$
$∴ P (X \leq 2) = c + 2 c = 2 c + c = 3 c = 3 \times \frac{1}{10} i . e P (X \leq 2) = \frac{3}{10}$

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