Photo AI
The random variable Y has a normal distribution with mean µ and standard deviation σ - Edexcel - A-Level Maths: Statistics - Question 3 - 2018 - Paper 1
Question 3
The random variable Y has a normal distribution with mean µ and standard deviation σ. The P(Y > 17) = 0.4. Find (a) P(µ < Y < 17) (b) P(µ - σ < Y < 17)
Worked Solution & Example Answer:The random variable Y has a normal distribution with mean µ and standard deviation σ - Edexcel - A-Level Maths: Statistics - Question 3 - 2018 - Paper 1
Step 1
P(µ < Y < 17)
Answer
To find the probability P(µ < Y < 17), we start with the given information that P(Y > 17) = 0.4.
Using the properties of the normal distribution, we know that:
Next, since the mean µ is where the distribution centers, we can standardize it. The area to the left of µ is also half of the total area, thus this gives:
Step 2
P(µ - σ < Y < 17)
Answer
For the second part, we need to calculate P(µ - σ < Y < 17).
We standardize the variable Y:
Let Z = (Y - µ) / σ.
Then,
( Y < 17 \implies Z < \frac{17 - µ}{σ} )
We need to evaluate ( P(Z < \frac{17 - µ}{σ}) ).
From the first part, we calculated that ( P(Y < 17) = 0.6 ).
Now, we will find P(Y < µ - σ):
Since P(Y > 17) = 0.4, this implies:
Calculating yields:
- Set ( P(Y < µ - σ) \approx 0.158 ) (from standard normal distribution table) joint with previous steps, we have: