A-Level Edexcel Maths Statistics Hypothesis Testing: The discrete random variable X h

The discrete random variable X has the following probability distribution | x | a | b | c | |---|---------|---------|---------| | P(X = x) | $\log_b a$ | $\log_b b$ | $\log_b c$ | 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_b-a$-|-$\log_b-b$-|-$\log_b-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... show full transcript

Worked Solution & Example Answer:The discrete random variable X has the following probability distribution | x | a | b | c | |---|---------|---------|---------| | P(X = x) | $\log_b a$ | $\log_b b$ | $\log_b c$ | 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

We start with the equation for the sum of probabilities:

$\log_b a + \log_b b + \log_b c = 1$

This implies that:

$\log_b (abc) = 1$

From logarithmic properties, we see that:

$abc = 36$

To find distinct integers for a, b, and c, we can consider the factors of 36, which can be represented as:

2, 3, and 6

Thus, we conclude that:

a = 2.

Step 2

Find (ii) the value of b

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Answer

Using the distinct integers identified, we have:

b = 3.

Step 3

Find (iii) the value of c

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Answer

Continuing with our chosen integers, we find:

c = 6.

Step 4

Find (b) P(X₁ = X₂)

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Answer

The probability that X₁ equals X₂ can be calculated by summing the squares of the individual probabilities:

$P(X_1 = X_2) = P(X = a)^2 + P(X = b)^2 + P(X = c)^2$

Calculating each:

$P(X = a) = \log_b (2)$
$P(X = b) = \log_b (3)$
$P(X = c) = \log_b (6)$

Using the previous results:

$P(X_1 = X_2) = (\log_3 2)^2 + (\log_3 3)^2 + (\log_3 6)^2$

Compute the specific probabilities and sum them to get:

P(X₁ = X₂) ≈ 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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