The discrete random variable $X$ has the following probability distribution, where $p$ and $q$ are constants - Edexcel - A-Level Maths Statistics - Question 2 - 2016 - Paper 1
Question 2
The discrete random variable $X$ has the following probability distribution, where $p$ and $q$ are constants.
| $x$ | -2 | -1 | 1/2 | 2 | 2 |
|------|----|----|-... show full transcript
Worked Solution & Example Answer:The discrete random variable $X$ has the following probability distribution, where $p$ and $q$ are constants - Edexcel - A-Level Maths Statistics - Question 2 - 2016 - Paper 1
Step 1
Write down an equation in $p$ and $q$
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Answer
The total probability must sum up to 1. So:
p+q+0.2+0.3+p=1
This can be simplified to:
2p+q+0.5=1
Thus, we have:
2p+q=0.5
Step 2
Given that $E(X) = 0.4$, find the value of $q$
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Answer
To find E(X), we calculate:
E(X)=(−2)p+(−1)q+(0.5)(0.2)+(2)(0.3)+(2)p
Substituting E(X)=0.4, we have:
E(X)=−2p−q+0.1+0.6+2p=0.4
This simplifies to:
p−q+0.7=0.4
Thus:
p−q=−0.3
Now, we have two equations:
2p+q=0.5
p−q=−0.3
Substituting equation (2) into equation (1) gives:
2p+(−0.3−p)=0.5
This leads us to find:
p=0.4
Then, substitute p=0.4 back into equation (2) to find q:
ightarrow q = 0.7$$.
Step 3
hence, find the value of $p$
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Answer
From the earlier calculation, we established:
p=0.4.
Step 4
Given also that $E(X^2) = 2.275$, find $Var(X)$
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To find the variance, we first need to calculate:
E(X2)=(−2)2imesp+(−1)2imesq+(0.5)2imes0.2+(2)2imes0.3+(2)2imesp
Substituting known values:
E(X2)=4p+q+0.05+1.2+4p
Which simplifies to:
5p+q+1.25=2.275
This can be rearranged to:
5p+q=1.025
Now we have:
2p+q=0.5
5p+q=1.025
Subtracting the first equation from the second gives:
ightarrow p = 0.175$$
Now substituting back to find $q$ gives:
$$2(0.175) + q = 0.5
ightarrow q = 0.15$$
Variance can then be calculated as:
$$Var(X) = E(X^2) - (E(X))^2 = 2.275 - (0.4)^2$$
Thus:
$$Var(X) = 2.275 - 0.16 = 2.115$$.
Step 5
Find $E(R)$
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Answer
To find E(R), we calculate:
E(R) = Eigg(\frac{1}{X}\bigg)
Using the probability distribution, this can be expressed as:
E(R)=P(X=−2)⋅−21+P(X=−1)⋅−11+P(X=0.5)⋅0.51+P(X=2)⋅21+P(X=2)⋅21
Substituting the probabilities we established earlier:
E(R)=p⋅−21+q⋅−11+0.2⋅0.51+0.3⋅21+p⋅21
This simplifies to:
ightarrow E(R) = 0.075$$.
Step 6
Find the probability that Sarah is the winner,
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Answer
For Sarah to win:
ightarrow X > \frac{1}{X}
ightarrow X^2 > 1$$
This implies:
$$X > 1 \text{ or } X < -1$$
Using the probabilities:
From the distribution:
$$P(X > 1) = P(X = 2) = 0.3\text{ and } P(X < -1) = P(X = -2) + P(X = -1) = p + q = 0.4$$
Thus, the total probability for Sarah winning is:
$$P(S > R) = 0.3 + 0.4 = 0.7.$
Step 7
Find the probability that Rebecca is the winner.
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Answer
For Rebecca to win:
ightarrow \frac{1}{X} > X
ightarrow 1 > X^2$$
This implies:
$$-1 < X < 1$$
Using the probabilities:
From the distribution:
$$P(-1 < X < 1) = P(X = -1) + P(X = 0.5) = q + 0.2$$
Substituting $q = 0.15, ext{ we find:}$
$$P(R > S) = 0.15 + 0.2 = 0.375.$
