Combining normal distributions | CAIE 9709 Probability & Statistics 2 Notes

Probability & Statistics 2

Mathematics9709

Build the new normal distribution explicitly

Let $D=aX+bY$ . If X and Y are independent normal variables, calculate these two quantities separately:

E(D)=a\mu_X+b\mu_Y

\operatorname{Var}(D)=a^2\sigma_X^2+b^2\sigma_Y^2

Then write $D\sim N(E(D),\operatorname{Var}(D))$ . In this notation, the second parameter of N is the variance.

Worked example

Current-paper normal difference

X and Y are independent with $X\sim N(44,16)$ and $Y\sim N(30,9)$ . Find $P(X-Y<15)$ .

X-Y\sim N(44-30,16+9)

X-Y\sim N(14,25)

z=\frac{15-14}{\sqrt{25}}=0.2

Answer

P(X-Y<15)=0.579

(9709/62/F/M/25 Q1)

Move a comparison onto one side

Make one normal variable

L < 3S

original event

move 3S left ↓

D = L − 3S < 0

E(D) = 0.1

Var(D) = 0.23

D is normal because L and S are independent normal variables.

Move all random terms to one side, then build the distribution of that side.

Worked example

Current-paper potato-bag comparison

Large and small bag masses are independent, with $L\sim N(2.5,0.05)$ and $S\sim N(0.8,0.02)$ . Find $P(L<3S)$ .

D=L-3S\sim N(0.1,0.23)

P(L<3S)=P(D<0)

z=\frac{0-0.1}{\sqrt{0.23}}=-0.209

Answer

P(L<3S)=0.417

(9709/61/O/N/25 Q6(b))

An unsigned difference needs two tails

Key idea

“The difference is more than k” usually means an absolute difference:

|X-Y|>k

. If the difference distribution is symmetric about zero, calculate one tail and double it.

Your turnIndependent absolute-difference check [5 marks]

Two independent completion times are each normally distributed with mean 125 minutes and variance 50. Find the probability that the difference between the two times is more than 12 minutes.

Show worked answer

D=X_1-X_2\sim N(0,100)

P(|D|>12)=2P(D>12)

z=12/10=1.2

2(1-\Phi(1.2))=0.230

Answer

P(|D|>12)=0.230

Common mistake

A difference subtracts means but adds independent variances. Do not write

\sigma_X^2-\sigma_Y^2

Build the new normal distribution explicitly

Let $D=aX+bY$ . If X and Y are independent normal variables, calculate these two quantities separately:

E(D)=a\mu_X+b\mu_Y

\operatorname{Var}(D)=a^2\sigma_X^2+b^2\sigma_Y^2

Then write $D\sim N(E(D),\operatorname{Var}(D))$ . In this notation, the second parameter of N is the variance.

Worked example

Current-paper normal difference

X and Y are independent with $X\sim N(44,16)$ and $Y\sim N(30,9)$ . Find $P(X-Y<15)$ .

X-Y\sim N(44-30,16+9)

X-Y\sim N(14,25)

z=\frac{15-14}{\sqrt{25}}=0.2

Answer

P(X-Y<15)=0.579

(9709/62/F/M/25 Q1)

Move a comparison onto one side

Make one normal variable

L < 3S

original event

move 3S left ↓

D = L − 3S < 0

E(D) = 0.1

Var(D) = 0.23

D is normal because L and S are independent normal variables.

Move all random terms to one side, then build the distribution of that side.

Worked example

Current-paper potato-bag comparison

Large and small bag masses are independent, with $L\sim N(2.5,0.05)$ and $S\sim N(0.8,0.02)$ . Find $P(L<3S)$ .

D=L-3S\sim N(0.1,0.23)

P(L<3S)=P(D<0)

z=\frac{0-0.1}{\sqrt{0.23}}=-0.209

Answer

P(L<3S)=0.417

(9709/61/O/N/25 Q6(b))

An unsigned difference needs two tails

Key idea

“The difference is more than k” usually means an absolute difference:

|X-Y|>k

. If the difference distribution is symmetric about zero, calculate one tail and double it.

Your turnIndependent absolute-difference check [5 marks]

Two independent completion times are each normally distributed with mean 125 minutes and variance 50. Find the probability that the difference between the two times is more than 12 minutes.

Show worked answer

D=X_1-X_2\sim N(0,100)

P(|D|>12)=2P(D>12)

z=12/10=1.2

2(1-\Phi(1.2))=0.230

Answer

P(|D|>12)=0.230

Common mistake

A difference subtracts means but adds independent variances. Do not write

\sigma_X^2-\sigma_Y^2