Question

Let X represent the weight in pounds of king salmon caught at the mouth of certain river and assume that X possesses a normal distribution with mean 30 and standard deviation 6. Calculate the probability that if a fisherman catches a salmon its weight will be:

a) At least 41 pounds
b) Between 20 and 40 pounds

Answer 1

Answer on Question #72466 - Math / Statistics and Probability

Let $X$ represent the weight in pounds of king salmon caught at the mouth of certain river and assume that $X$ possesses a normal distribution with mean 30 and standard deviation 6. Calculate the probability that if a fisherman catches a salmon its weight will be:

a) At least 41 pounds

b) Between 20 and 40 pounds

Solution:

The distribution function $X$ has the form

$$F_{X}(x) = \frac{1}{\sqrt{2\pi} \cdot 6} \int_{-\infty}^{x} e^{-\frac{(s - 30)^2}{72}} ds,$$

and

$$P\{a \leq X \leq b\} = F_{X}(b) - F_{X}(a).$$

a) $P\{X \geq 41\} = 1 - F_{X}(41) = 0.033 = 3.3\%$

b) $P\{20 \leq X \leq 40\} = F_{X}(40) - F_{X}(20) = 0.904 = 90.4\%$

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