The discrete random variable X has the following probability distribution, where p and q are constants. | x | -2 | -1 | 1/2 | 3/2 | 2 | | P(X = x) | p | q | 0.2 | 0.3 | p | (a) Write down an equation in p and q Given that E(X) = 0.4 (b) find the value of q (c) hence find the value of p Given also that E(X^2) = 2.275 (d) find Var(X) Sarah and Rebecca play a game. A computer selects a single value of X using the probability distribution above. Sarah's score is given by the random variable S = X and Rebecca's score is given by the random variable R = 1/X (e) Find E(R) Sarah and Rebecca work out their scores and the person with the higher score is the winner. If the scores are the same, the game is a draw. (f) Find the probability that (i) Sarah is the winner, (ii) Rebecca is the winner.

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

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

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The discrete random variable X has the following probability distribution, where p and q are constants. | x | -2 | -1 | 1/2 | 3/2 | 2 | | P(X = x) | p | q | 0.2 | 0... 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

Since the total probability must equal 1, we have the equation:

$p + q + 0.2 + 0.3 + p = 1$

This simplifies to: $2p + q + 0.5 = 1$

Thus, $2p + q = 0.5$

Step 2

find the value of q

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Answer

We know that $E(X) = -2p - q - 0.5 + 1.5p + 2p$ Substituting E(X) = 0.4, we get:

\Rightarrow -p - q + 1.5 = 0.4\ \Rightarrow -p - q = -1.1\ \Rightarrow p + q = 1.1$$ Now we have two equations: 1. $$2p + q = 0.5$$ 2. $$p + q = 1.1$$ Subtracting these two equations: $$p = 0.15\ \Rightarrow q = 1.1 - 0.15 = 0.95$$

Step 3

hence find the value of p

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Answer

Using the value of q found in the previous step, we substitute back into the first equation:

\Rightarrow 2p = 0.5 - 0.95 = -0.45\ \Rightarrow p = -0.225$$

Step 4

find Var(X)

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Answer

Using the given value for E(X^2) = 2.275, we can find Var(X) as:

Var(X) = 2.275 - (0.4)^2 = 2.275 - 0.16 = 2.115$$

Step 5

Find E(R)

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Answer

The expectation E(R) can be computed as follows:

$E(R) = E\left(\frac{1}{X}\right) = P(X = -2) * \frac{1}{-2} + P(X = -1) * \frac{1}{-1} + P(X = 0.5) * 2 + P(X = 1.5) * \frac{2}{3} + P(X = 2) * \frac{1}{2}$ Substituting, we get:

$= p(-\frac{1}{2}) + q(-1) + 0.2(2) + 0.3(\frac{2}{3}) + p(\frac{1}{2})$ Substituting p and q, we find:

$E(R) = -\frac{0.15}{2} - 0.95 + 0.4 + 0.2 + 0.075 = 0.45$

Step 6

Find the probability that Sarah is the winner

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Answer

The probability that Sarah wins can be computed using the given values of X:

= P(S > \frac{1}{X})$$ We can integrate over all values of X based on the distribution of X to calculate this probability.

Step 7

Find the probability that Rebecca is the winner

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Answer

The probability that Rebecca wins can be calculated similarly:

= P(\frac{1}{X} > X)$$ Again, we integrate over the distribution of X to find this.

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

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

Write down an equation in p and q

find the value of q

hence find the value of p

find Var(X)

Find E(R)

Find the probability that Sarah is the winner

Find the probability that Rebecca is the winner

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