The discrete random variable $X$ has the following probability distribution
| $x$ | $a$ | $b$ | $c$ |
| P($X = x$) | $ ext{log}_b a$ | $ ext{log}_b b$ | $ ext{log}_b c$ |
where
- $a$, $b$ and $c$ are distinct integers ($a < b < c$)
- all the probabilities are greater than zero
(a)
(i) the value of $a$
(ii) the value of $b$
(iii) the value of $c$
Show your working clearly - Edexcel - A-Level Maths Statistics - Question 6 - 2021 - Paper 1
Question 6
The discrete random variable $X$ has the following probability distribution
| $x$ | $a$ | $b$ | $c$ |
| P($X = x... show full transcript
Worked Solution & Example Answer:The discrete random variable $X$ has the following probability distribution
| $x$ | $a$ | $b$ | $c$ |
| P($X = x$) | $ ext{log}_b a$ | $ ext{log}_b b$ | $ ext{log}_b c$ |
where
- $a$, $b$ and $c$ are distinct integers ($a < b < c$)
- all the probabilities are greater than zero
(a)
(i) the value of $a$
(ii) the value of $b$
(iii) the value of $c$
Show your working clearly - Edexcel - A-Level Maths Statistics - Question 6 - 2021 - Paper 1
Step 1
Find (i) the value of a
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Answer
To find the value of a, we start with the equation that the sum of the probabilities equals 1:
extlogba+extlogbb+extlogbc=1.
Using the properties of logarithms, this can be rewritten as:
extlogb(abc)=1.
This implies:
abc=b1=b.
Next, we recognize that ab=36 since abc=36. Given that a, b, and c are distinct integers greater than 1, we can conclude:
a=2,b=3,c=6.
Step 2
Find (ii) the value of b
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Answer
As concluded previously, the value of b is 3. This can be seen directly from the previously established equations involving distinct integers a, b, and c.
Step 3
Find (iii) the value of c
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Answer
From the earlier calculations, we determined that c is equal to 6, following from the relationship of the factors of 36 such that a=2, b=3, and c=6.
Step 4
Find P(X1 = X2)
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Answer
For independent random variables X1 and X2, which each follow the same distribution as X, the probability can be calculated as:
P(X1=X2)=P(X=a)2+P(X=b)2+P(X=c)2
Substituting in the values, we have:
P(X1=X2)=(extlogb2)2+(extlogb3)2+(extlogb6)2.
Calculating this gives:
P(X1=X2)extevaluatestoapproximately0.381.
