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- Below is the probability density function for the distance, in inches, that a dart thrown by your professor lands from the “bullseye” of a dart board. The equation of the curve is: d(x) = 1/4 − 1/32 x a.What’s the probability that a dart will land less that 8 inches from the bullseye? b.What’s the probability that a dart will land between 2 and 4 inches from the bullseye? c.What’s the probability that a dart will land exactly 2 inches from the bullseye? Ex- plain. d.What’s the probability that a dart will land less than 6 inches away from the bullseye?The time between infection and the display of symptoms for streptococcal sore throat is a random variable whose probability density function can be approximated by the following. f(t) = 1 15,676 t2e−0.05t if 0 ≤ t ≤ 150 and f(t) = 0 otherwise (t measured in hours) (a) What is the probability that an infected patient will display symptoms within the first 72 hours? (Round your answer to three decimal places.) (b) What is the probability that an infected patient will not display symptoms until after 60 hours? (Round your answer to three decimal places.)Martian potatoes begin to sprout very quickly after planting. Suppose X is the number of days after planting until a Martian potato sprouts. Then X has the following probability density function: f(x)= 2/7e−x + 3/14e−x/2 + 1/14e−x/4 for 0 ≤ x ≤ ∞ and 0 otherwise. a)What is the probability that a Martian potato takes no more than 4 days to sprout? b) What is the probability that X >4? c) What is the probability that 2< X < 4? d) What is the expected value of X (E(X))? e) What is the expected value of X2 ? f) What is the variance of X? g) What is the standard deviation of X? h) What is the probability that X is more than 2 standard deviations above its expected value? i) What is the expected value of X4 ? j) What is the probability that X is within 1 standard deviation of its expected value? k) What is the probability that X = .6?