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

The discrete random variable X can take only the values 2, 3 or 4 - Edexcel - A-Level Maths Statistics - Question 6 - 2008 - Paper 2

Question 6

The discrete random variable X can take only the values 2, 3 or 4. For these values the cumulative distribution function is defined by $$F(x) = \frac{(x+k)^2}{25}... show full transcript

Worked Solution & Example Answer:The discrete random variable X can take only the values 2, 3 or 4 - Edexcel - A-Level Maths Statistics - Question 6 - 2008 - Paper 2

Step 1

Find k.

96%

114 rated

Answer

To find the value of $k$ , we first need to ensure that the cumulative distribution function $F(x)$ satisfies the property that $F(4) = 1$ when $x = 4$ :

$F(4) = \frac{(4+k)^2}{25} = 1$

This gives us:

$(4+k)^2 = 25.$

Taking the square root of both sides:

$4+k = 5 \quad \text{or} \quad 4+k = -5$

Since $k$ is a positive integer, we discard $4+k = -5$ and solve:

$4+k = 5 \Rightarrow k = 1.$

Thus, we find that $k = 1$ .

Step 2

Find the probability distribution of X.

99%

104 rated

Answer

Now, we can use the value of $k$ to find the probability distribution of $X$ .

Using the cumulative distribution function:

For $x = 2$ :
$F(2) = \frac{(2+1)^2}{25} = \frac{3^2}{25} = \frac{9}{25}.$
Thus,
$P(X=2) = F(2) = \frac{9}{25}.$
For $x = 3$ :
$F(3) = \frac{(3+1)^2}{25} = \frac{4^2}{25} = \frac{16}{25}.$
Thus,
$P(X=3) = F(3) - F(2) = \frac{16}{25} - \frac{9}{25} = \frac{7}{25}.$
For $x = 4$ :
$F(4) = \frac{(4+1)^2}{25} = \frac{5^2}{25} = 1.$
Thus,
$P(X=4) = F(4) - F(3) = 1 - \frac{16}{25} = \frac{9}{25}.$

Finally, we summarize the probability distribution of $X$ :

x	P(X=x)
2	9/25
3	7/25
4	9/25

The discrete random variable X can take only the values 2, 3 or 4 - Edexcel - A-Level Maths Statistics - Question 6 - 2008 - Paper 2

Worked Solution & Example Answer:The discrete random variable X can take only the values 2, 3 or 4 - Edexcel - A-Level Maths Statistics - Question 6 - 2008 - Paper 2

Find k.

Find the probability distribution of X.

Join the A-Level students using SimpleStudy...