A-Level Edexcel Maths Statistics Basic Probability: The discrete random variable X h

The discrete random variable X has the following probability distribution | x | a | b | c | | P(X = x) | log_s(a) | log_s(b)| log_s(c) | where - a, b and c are distinct integers (a < b < c) - all the probabilities are greater than zero (a) Find (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)-|-log_s(a)----|-log_s(b)|-log_s(c)--|--where----a,-b-and-c-are-distinct-integers-(a-<-b-<-c)---all-the-probabilities-are-greater-than-zero--(a)-Find--(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.png

The discrete random variable X has the following probability distribution | x | a | b | c | | P(X = x) | log_s(a) | log_s(b)| l... show full transcript

Worked Solution & Example Answer:The discrete random variable X has the following probability distribution | x | a | b | c | | P(X = x) | log_s(a) | log_s(b)| log_s(c) | where - a, b and c are distinct integers (a < b < c) - all the probabilities are greater than zero (a) Find (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 for the total probability:

$ext{log}_s(a) + ext{log}_s(b) + ext{log}_s(c) = 1$

Using the properties of logarithms, this can be expressed as:

$ext{log}_s(abc) = 1$

Thus, we have:

$abc = s^1 = s$

Next, knowing that all probabilities must be greater than zero, we establish:

$36 - 2 imes 3 = abc = 36$

Thus, from the product of distinct integers a, b, and c, we find:

Possible values include (2, 3, 6).

So, the value of a = 2.

Step 2

Find (ii) the value of b

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Answer

Using the sequence of distinct integers (2, 3, 6): Since a = 2,

The next integer in the sequence following a would be:

b = 3.

Step 3

Find (iii) the value of c

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Answer

Continuing from the previous values:

Since a = 2 and b = 3,
The largest integer c should be:
c = 6.

Step 4

Find (b) P(X₁ = X₂)

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Answer

To find P(X₁ = X₂), we use the probabilities:

$P(X₁ = X₂) = P(X = a)^2 + P(X = b)^2 + P(X = c)^2$

Substituting the values gives:

For P(X = a) = log_s(2)
For P(X = b) = log_s(3)
For P(X = c) = log_s(6)

Calculating:

does yield values:

Combining: $P(X₁ = X₂) = ( ext{log}_s(2))^2 + ( ext{log}_s(3))^2 + ( ext{log}_s(6))^2$

Approximating yields: $ext{P}(X_1 = X_2) = 0.381.$

Find (i) the value of a

Find (ii) the value of b

Find (iii) the value of c

Find (b) P(X₁ = X₂)

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