Question 5

Consider the mathematical statement: $X \sim N(70,100)$.
(a) Explain the meaning of the above statement.

Find
(b) $P(X> 50)$
(c) $P(X \leq 60)$
(d) $P(X> 90)$
(e) $P(60 \leq X \leq 80)$

Step 1 ：The mathematical statement $X \sim N(70,100)$ means that X follows a normal distribution with a mean of 70 and a variance of 100.

Step 2 ：The standard deviation is the square root of the variance, so the standard deviation is 10.

Step 3 ：To find $P(X>50)$, we need to calculate the cumulative distribution function (CDF) at 50 and subtract it from 1. This is because the CDF at a point x gives the probability that X is less than or equal to x. So, $1 - CDF(50)$ gives the probability that X is greater than 50.

Step 4 ：Using the mean of 70 and standard deviation of 10, we find that $P(X>50) \approx 0.977$.

Step 5 ：Final Answer: $P(X>50) \approx \boxed{0.977}$