Solution — 2025 Question 10 Normal Distribution

(a)

Let $R, G$ and $Y$ be the random variables representing the masses of red, green and yellow peppers respectively

\begin{matrix} R \sim N (230, 35^{2}) \\ G \sim N (225, 30^{2}) \\ Y \sim N (220, 32^{2}) \end{matrix}

\begin{matrix} R_{1} + R_{2} & \sim N (2 (230), 2 {(35)}^{2}) \\ R_{1} + R_{2} & \sim N (460, 2450) \end{matrix}

\begin{matrix} (G + Y) & \sim N (225 + 220, 30^{2} + 32^{2}) \\ (G + Y) & \sim N (445, 1924) \end{matrix}

\begin{matrix} R_{1} + R_{2} - (G + Y) & \sim N (460 - 445, 2450 + 1924) \\ R_{1} + R_{2} - (G + Y) & \sim N (15, 4374) \end{matrix}

$\begin{matrix} P (R_{1} + R_{2} > G + Y) & = P (R_{1} + R_{2} - (G + Y) > 0) \\ = 0.589 71 \\ = 0.590 ∎ (to 3 s.f.) \end{matrix}$

[finding \mathrm{P}\left(R1+ R2 - (G+Y) > 0\right)]

(b)

\begin{matrix} 1.2 Y & \sim N (1.2 (220), {1.2}^{2} {(32)}^{2}) \\ 1.2 Y & \sim N (264, 1474.6) \end{matrix}

\begin{matrix} R - 1.2 Y & \sim N (230 - 264, 35^{2} + 1474.6) \\ R - 1.2 Y & \sim N (- 34, 2699.6) \end{matrix}

$\begin{matrix} P (R > 1.2 Y) & = P (R - 1.2 Y \geq 0) \\ = 0.256 43 \\ = 0.256 ∎ (to 3 s.f.) \end{matrix}$

[finding \mathrm{P}\left(R - 1.2Y \geq 0\right)]

(c)

$\begin{matrix} P (R < 220) & = 0.387 55 \end{matrix}$

[finding \mathrm{P}\left(R < 220\right)]

$\begin{matrix} P (G < 220) & = 0.433 82 \end{matrix}$

[finding \mathrm{P}\left(G < 220\right)]

$\begin{matrix} P (Y < 220) & = 0.5 \end{matrix}$

[finding \mathrm{P}\left(Y < 220\right)]

$\begin{matrix} P (yellow | mass less than 220) \\ = \frac{P (yellow \cap mass less than 220)}{P (mass less than 220)} \\ = \frac{0.2 (0.5)}{0.5 (0.387 55) + 0.3 (0.433 82) + 0.2 (0.5)} \\ = 0.235 89 \\ = 0.236 ∎ (to 3 s.f.) \end{matrix}$

(d)

$\begin{matrix} P (G > 220) & = 0.566 18 \end{matrix}$

[finding \mathrm{P}\left(G > 220\right)]

Let $X$ represent the number of green peppers in a bag of $20$ peppers that has mass at least 220 grams

$ ParseError: Can't use function '$' in math mode at position 1: $̲ X \sim \mathrm{B} \left( 20, 0.566,18 \right)

\begin{align*} \mathrm{P}\left(X \geq 15\right) &= 1 - \mathrm{P}\left(X \leq 14\right) \\ &= 1 - 0.926\,61 \\ &= 0.073\,394 \\ &= 0.0734\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[probability of X \geq 15]