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- (3) Let X = b >(8,-) find E(5+6x) and distribution function.Suppose X = time to wait for a cab and you know you're going to wait at least 6 minutes, but no more than 10 minutes. Suppose X has a uniform distribution. What is f(x)? f(x) = 1/6 f(x) = 0 f(x) = 1/10 O f(x) = 1/4Use the geometric distribution to derive E(Y), E(Y^2), and V(Y) from the Poisson distribution
- Derive the formulas for the mean and the varianceof the Poisson distribution by first evaluating E(X) andE[X(X − 1)].The geometric distribution gives the probability that the first success occurs at the xth trial with success probability p. f(x)=(1-p)x-1p, x=1,2,3,... Show that E(X)=1/pThe failure time (T) of an electronic circuit board follows exponentially distribution with failure rate λ = 10−4/h. What is the probability that (i) it will fail before 1000 h? (ii) it will survive at least 10,000 h ? (iii) it will fail between 1000 and 10,000 h? (iv)Determine the mean time to failure. .
- Find P(X=4) if X has a Poisson distribution such that 3P(X=1)=P(X=2)C1. Let X be a continuous random variable with PDF f(x) = (2-x) for -1 ≤ x ≤ c and f(x) = 0 otherwise. (a) Explaining your work, find the value of the constant c. (b) What is P(X > 1)? (c) Calculate the expectation of X. (d) Calculate the variance of X.C1. Let X be a continuous random variable with PDF f(x) = (2-x) ² for -1 1)? (c) Calculate the expectation of X. (d) Calculate the variance of X.