Question

9] Suppose X is Normal random variable with mean 10 and SD 4. What is the probability that X is between 7 and 13? [10] Suppose X is Normal random variable with unknown mean mu and SD 4. For what value of mu, 80th percentile of X will be 23? (11] Lifetime of Carbon 14, X, is modeled by Exponential distribution with mean of 8223.7 years Determine its half-life (50th percentile).

Answer 1

9)

Given

$\mu$ = 10, $\sigma$ = 4

We convert this to standard normal as

P (X < x) = P( Z < x - $\mu$ / $\sigma$)

So,

P( 7 < X < 13) = P (X < 13) - P( X < 7)

= P (Z < 13 - 10 / 4) - P (Z < 7 - 10 / 4)

= P(Z < 0.75) - P( Z < -0.75)

= 0.7734 - 0.2266

= 0.5468

10)

80th percentile = $\mu$ + Z$\sigma$ ,Where Z is critical value at 80% confidence level.

23 = $\mu$ + 0.8416 * 4

Solve for $\mu$

$\mu$ = 19.6336