Question

Let X be a normal random variable with its mean equal to 65 and
standard deviation equal to 12. Find the probabilities for normal
distribution.

1)p(x>48)

2)p(35<x<43)

3)p(x<37)

Answer 1

Answer on Question #64236 – Math – Statistics and Probability Question

Let $X$ be a normal random variable with its mean equal to 65 and standard deviation equal to 12. Find the probabilities for normal distribution.

1) $P(X > 48)$

2) $P(35 < X < 43)$

3) $P(X < 37)$

Solution

Let $\xi$ be a standard normal random variable.

Then

1) If $E(X) = 65$, $sd(X) = 12$, then

$$P(X > 48) = P\left(\xi > \frac{48 - 65}{12}\right) = P(\xi > -1.42) = 1 - \Phi(-1.42) = 1 - 0.0778 = 0.9222.$$

Here $\Phi(z)$ is the standard normal cumulative distribution function of $\xi$. The value of $\Phi(z)$ can be found using statistical tables

(for example, see https://homes.cs.washington.edu/~jrl/normal_cdf.pdf).

2) $P(35 < X < 43) = P\left(\frac{35 - 65}{12} < \xi < \frac{43 - 65}{12}\right) = P(-2.5 < \xi < -1.83) =$

$$= \Phi(-1.83) - \Phi(-2.5) = 0.0336 - 0.0062 = 0.0274.$$

3) $P(X < 37) = P\left(\xi < \frac{37 - 65}{12}\right) = P(\xi < -2.33) = \Phi(-2.33) = 0.0099.$

Answer:

1) $P(X > 48) = 0.9222$.

2) $P(35 < X < 43) = 0.0274$.

3) $P(X < 37) = 0.0099$.

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