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Five automobiles of the same type are to be driven on a 300-mile trip. The first two will use an economy brand of gasoline, and the other three will use a premium brand. Let X1, X2, X3, X4 and X5 be random variables representing the fuel efficiencies (mpg) for the five cars. Suppose these variables are independent and normally distributed with mu1 = mu2 = 20, mu3 = mu4 = mu5 = 21, and sigma 2 = 4 for the economy brand and 3.5 for premium brand. Define a random variable Y by Y = X1 + X2 /2 - X3 + X4 + X5 / 3, so that Y is a measure of the difference in efficiency between economy gas and premium gas. Compute P(0 le Y) and P(-1 le Y le 1). [Hint: write Y = c1X1 + ... +c5X5.]

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temmydbrain

Answer:

Step-by-step explanation:

A random variable, which is a variable that has an unknown value or a function that allocates values to each of an experiment's outcomes.

solving this question;

Y is a normal distribution with:

mean = (20+20)/2 - (21+21+21)/3 = 20-21 = -1

and

standard deviation = sqrt((4+4)/2^2 + (3.5+3.5+3.5)/3^2) = 1.7759

Therefore:

P( 0 <= Y ) =

P((0-(-1))/1.7759 < Z) = P(0.56 < Z) = 1 -0.7123 = 0.2877

and

P(-1 <= Y <= 1) =

P((-1+1)/1.7759 <= Z <= (1+1)/1.7759) =

P(0 <= Z <= 1.13) =

0.8708 - 0.5 =

0.3708

yoodyannapolis

[tex]P( 0 \leq Y )= 0.2877, P( -1 \leq Y \leq 1)= 0.3708[/tex]

We have random variable [tex]Y =\frac{X_1 + X_2 - X_3 + X_4 + X_5}{3}[/tex]

A random variable is a variable that has an unknown value or a function that allocates values to each of an experiment's outcomes.

 Y is a normal distribution with

Mean

[tex]=\frac{(20+20)}{2} - \frac{(21+21+21)}{3} \\= 20-21 \\= -1[/tex]

Standard deviation

[tex]= \sqrt{(\frac{(4+4)}{2^2}) +\frac{ (3.5+3.5+3.5)}{3^2} )} \\= 1.7759[/tex]

 So,

[tex]P( 0 \leq Y ) \\=P(\frac{(0-(-1))}{1.7759} < Z)\\= P(0.56 < Z) \\= 1 -0.7123\\ P( 0 \leq Y )= 0.2877[/tex]

And

[tex]P(-1 \leq Y \leq 1) \\=P(\frac{(-1+1)}{1.7759} \leq Z \leq \frac{(-1+1)}{1.7759})\\=P(0 \leq Z \leq 1.13)\\ = 0.8708 - 0.5\\ =0.3708[/tex]

Therefore, [tex]P( 0 \leq Y )= 0.2877, P( -1 \leq Y \leq 1)= 0.3708[/tex]

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