The score when a spinner is spun is given by the discrete random variable $X$ with the following probability distribution, where $a$ and $b$ are probabilities - Edexcel - A-Level Maths Statistics - Question 5 - 2018 - Paper 1
Question 5
The score when a spinner is spun is given by the discrete random variable $X$ with the following probability distribution, where $a$ and $b$ are probabilities.
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Worked Solution & Example Answer:The score when a spinner is spun is given by the discrete random variable $X$ with the following probability distribution, where $a$ and $b$ are probabilities - Edexcel - A-Level Maths Statistics - Question 5 - 2018 - Paper 1
Step 1
Explain why E(X) = 2
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Answer
The expected value E(X) of a random variable can be calculated using the formula:
ightarrow 4a - b = 0$$
Thus, we can see that when we rearrange and account for the symmetry, it becomes clear why $E(X) = 2$. Therefore, $E(X)$ indeed equals 2.
Step 2
Find a linear equation in a and b.
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Answer
From the uniform probability distribution, we know that:
ightarrow 4a + 2b = 1$$
This simplifies into:
$$2a + b = 0.5$$
This is the first linear equation in terms of $a$ and $b$.
Step 3
Find a second equation in a and b and simplify your answer.
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Answer
Using the variance equation:
Var(X)=E(X2)−(E(X))2
To calculate E(X2), we have:
E(X2)=(−1)2b+02a+22a+42a+52b
This gives us:
E(X2)=b+0+4a+16a+25b=26b+20a
Since we know that Var(X)=7.1, we can become:
ightarrow (26b + 20a) - 4 = 7.1$$
This rearranges to:
$$26b + 20a = 11.1$$
This would be the second equation.
Step 4
Solve your two equations to find the value of a and the value of b.
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Answer
Now we have the system of equations:
2a+b=0.5
26b+20a=11.1
From equation 1, we can isolate b:
b=0.5−2a
Substituting into equation 2:
26(0.5−2a)+20a=11.1
Expanding gives:
13−52a+20a=11.1
Rearranging leads to:
ightarrow a = 0.059375$$
Using $a$ in equation 1 to find $b$:
$$b = 0.5 - 2(0.059375) = 0.38125$$
Thus, we find:
$$a = 0.059375, b = 0.38125$$.
Step 5
Find E(Y).
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Answer
To find E(Y) where Y=10−3X, we use the transformation property of expected values:
E(Y)=E(10−3X)=10−3E(X)
Substituting E(X)=2 gives:
E(Y)=10−3(2)=10−6=4.
Step 6
Find Var(Y).
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Answer
To find P(Y>X), we first substitute:
ightarrow 10 - 3X > X
ightarrow 10 > 4X
ightarrow X < 2.5$$
Thus, we need the probability of $X$ being less than 2.5. Since $X$ can take values of -1, 0, and 2, we find:
$$P(Y > X) = P(X = -1) + P(X = 0) + P(X = 2) = b + a + a = b + 2a$$
Using our values from earlier, we compute:
$$P(Y > X) = 0.38125 + 2 imes 0.059375 = 0.5$$.
