If f(x) is a probability density on the real line, then which of the following is NOT a valid probability density?

Let X1, X2, ... be a sequence of independent and identically distributed random variables having a Binomial(3, 14) distribution. For j = 1, 2, ..., define Yj = 1 if Xj ≤ √5, and Yj = 0 otherwise. If 1n ∑j=1n Yj2 converges almost surely to a constant c as n → ∞, then the value of c is:

Let X be a single sample from an absolutely continuous distribution with probability density function f(x|θ) = 2θ2 (θ - x) for 0 0 is unknown. Which of the following intervals is a 95% confidence interval for θ?

Let X | θ ~ Uniform[0, θ] and let θ have an Exponential distribution with mean λ, where λ > 3 is known. If the realized value of X is 2025, then the posterior mode equals:

Let {Yn : n ≥ 1} be a sequence of independent and identically distributed random variables, where Yi ~ Bernoulli(12). Define Z = ∑n=1∞ 4 Yn5n. Then, which of the following statements is true?

Let X1 and X2 be a random sample from a Uniform[0, θ] distribution, where θ > 0. For testing H0 : θ = 1 against H1 : θ = 2, consider a test which rejects H0 if X1 + X2 > 45. What is the probability of Type-I error?

Which probability distribution model governs the fluctuations in the fraction defective in quality control sampling?

Consider the following probability mass function of a random variable X: P(X = k) = 14 when x = 0, p when x = 1, and 0 otherwise. If p = 0.25, then the mean of X is:

The variance of a uniformly distributed random variable of length 2 is: