12. If a random variable X has a discrete distribution for which the p.f. is f(x), then the value of x for which f(x) is maximum is called the mode of the distribution. If this same maximum f(x) is attained at more than one value of x, then all such values of x are called modes of the distribution. Find the mode or modes of the binomial distribution with parameters n and p. Hint: Study the ratio f(x+1|n, p)/f(x\n, p).

Suppose that X has a Poisson distribution with h = 80. Find theprobability P(x is less than or equal to 80)

In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Poisson distribution with probability mass function f(x), where f(x) = θ x e −θ/x! and θ is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was.

Suppose that a random variable X has a hypergeometric distribution with parameters n = 36, N = 50, and M = 28. What is smallest possible value for X?   What is largest possible value for X?

Suppose that X is a discrete random variable with P(X = 0) = 0.3, P(X = 1) = 0.2, P(X =2) = 0.15, and P(X = 3) = 0.35. Graph the frequency function and the cumulative distribution function of X.

A company has 9000 arrivals of Internet traffic over a period of 18,050 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= (μ^x • e^−μ) / x! to find the probability of exactly 2 arrivals in one thousandth of a​ minute, what are the values of μ​, ​x, and e that would be used in that​ formula?

If np is greater than or equal to 5 and nq is greater than or equal to 5, estimate P(at least 7) n is equal to 13 and p is equal to 0.6 by using the normal distribution as an approximation to the binomial distribution; if np is less than 5 or nq is less than 5, then state that the normal approximation is not suitable

Suppose that you have ten lightbulbs, that the lifetime ofeach is independent of all the other lifetimes, and that eachlifetime has an exponential distribution with parameter l.a. What is the probability that all ten bulbs fail beforetime t?b. What is the probability that exactly k of the ten bulbsfail before time t?c. Suppose that nine of the bulbs have lifetimes that areexponentially distributed with parameter l and thatthe remaining bulb has a lifetime that is exponentiallydistributed with parameter u (it is made byanother manufacturer). What is the probability thatexactly five of the ten bulbs fail before time t?