What is the smallest value of k in Chebyshev’s theo-rem for which the probability that a random variable will take on a value between μ − kσ and μ + kσ is(a) at least 0.95;(b) at least 0.99?

8. The time (in minutes) that it takes a car mechanic to replace a car battery is a random variable X that follows an exponential distribution with mean 15 minutes. a) Find P(14 < X < 22). Round off your answer to two decimal places. x 22 1 P(14 < X <22) = √₂/1² 15 e 15dx = P(x < 22) — P(x < 14) x 15 122 = e 15 22 1 - e 14 or 0.8 = = 22 14 = −e 15 + e15 = 0.16254≈ 0.16 - (1 – b) Find the 80th percentile. Round off to the nearest whole number. 1 X S 15 14 e 15 t e 15 dt X 0.8 = 1 e 15 X e 15 = = 0.2 14 22 = e 15 e 15 = 0.16 x — = ln (0.2). This gives x = 24.14156 15

If x is a binomial random variable, compute p(x) for each of the following cases:   (a) n = 6, x = 2, p = 0.9 p(x) =  (b) n = 6, x = 1, p = 0.3