Let X and Y be two independent Binomial random variables such that X B(2, p) and Y B(4, p) . If P(X ≥ 1) = 5/9 , find (i) P(Y = 4) and (ii) V(2X - Y) - Mathematics MCQ Mathematics Mock Test - 8 2026 | ExamTest.live

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Let X and Y be two independent Binomial random variables such that X \sim B (2, p) and Y \sim B (4, p) . If P (X \geq 1) = 5/9, find (i) P (Y = 4) and (ii) V (2 X - Y) .

Solution & Step-by-step Explanation

For X \sim B (2, p) : P (X \geq 1) = 1 - P (X = 0) = 1 - (1 - p)^{2} .Given 1 - (1 - p)^{2} = 5/9 ⟹ (1 - p)^{2} = 4/9 ⟹ 1 - p = 2/3 ⟹ p = 1/3 .Then q = 2/3 .(i) For Y \sim B (4, 1/3) : P (Y = 4) = (4 4) p^{4} q^{0} = (1/3)^{4} = 1/81 (ii) Variance V (2 X - Y) :Since X, Y are independent: V (2 X - Y) = 2^{2} V (X) + (- 1)^{2} V (Y) = 4 V (X) + V (Y) . V (X) = n_{x} pq = 2 \cdot (1/3) \cdot (2/3) = 4/9 V (Y) = n_{y} pq = 4 \cdot (1/3) \cdot (2/3) = 8/9 $ V (2 X - Y) = 4 (4/9) + 8/9 = 16/9 + 8/9 = 24/9 $

Try it yourself before checking the explanation above.

Let X and Y be two independent Binomial random variables such that X \sim B (2, p) and Y \sim B (4, p) . If P (X \geq 1) = 5/9, find (i) P (Y = 4) and (ii) V (2 X - Y) .

\frac{1}{27}, \frac{24}{9}

\frac{1}{81}, \frac{24}{9}

\frac{1}{81}, \frac{25}{9}

None

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