Let X be the discrete random variable representing the number x appeared on the face of a biased die when it is rolled. The probability distribution of X is X = x 1 2 3 4 5 6 P X = x 0 . 1 0 . 15 0 . 3 0 . 25 k k Then variance of X is

Question

Let X be the discrete random variable representing the number x appeared on the face of a biased die when it is rolled. The probability distribution of X is X = x 1 2 3 4 5 6 P X = x 0 . 1 0 . 15 0 . 3 0 . 25 k k Then variance of X is, 1 . 64, 1 . 94, 2 . 16, 2 . 28

MARKS App · Accepted Answer

We know that, ∑ P i = 1 ⇒ 2 k + 0 . 8 = 1 ⇒ k = 0 . 1 x i 2 X = x i P X = x P i x i P i x i 2 1 1 0.1 0.1 0.1 4 2 0.15 0.30 0.6 9 3 0.3 0.9 2.7 16 4 0.25 1.0 4 25 5 0.1 0.5 2.5 36 6 0.1 0.6 3.6 ∑ P i x i = 3 . 4 ∑ P i x i 2 = 13 . 5 Variance x = ∑ P i x i 2 - ∑ P i x i 2 = 13 . 5 - 3 . 4 2 = 13 . 5 - 11 . 56 = 1 . 94

$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P (X = x)$	$0.1$	$0.15$	$0.3$	$0.25$	$k$	$k$

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$x_{i}^{2}$	$X = x_{i}$	$P (X = x)$	$P_{i} x_{i}$	$P_{i} x_{i}^{2}$
1	1	0.1	0.1	0.1
4	2	0.15	0.30	0.6
9	3	0.3	0.9	2.7
16	4	0.25	1.0	4
25	5	0.1	0.5	2.5
36	6	0.1	0.6	3.6
			$\sum P_{i} x_{i} = 3.4$	$\sum P_{i} x_{i}^{2} = 13.5$