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- 4. The battery life of a fully charge iPhone 13 Pro on a standby follows a normally distributed with mean 7 hour and variance o² fl = 1 hours. = (a) Write out the integral form of the probability that a fully-charged iPhone 13 Pro last between 7 and 9 hours on standby. then use the pnorm function in R to find this probability. (b) What is the probability that a fully-charged iPhone 13 Pro last less than 5 hours? (c) What is the probability that a fully-charged iPhone 13 Pro last more than 9 hours? (d) What is the minimum time a fully-charged iPhone 13 Pro need to last in order to be in the top 10% best quality?Let P and Q be two independent Random variables with Var(P) = 5 and Var(Q) = 4. Find Var(7P+4Q-1).Derive the variance of the Student's t distribution using the definition Var (x) = ELX2] - EX)² .
- The amount of time a cashier spends cashing out each customer at a store is modeled by an exponential random variable with parameter λ =1/3 minutes. Assume that the time the cashier spends with each customer is independent and identically distributed. Approximate the probability that the cashier will spend more than sixty minutes cashing out a total of eighteen customers. You may leave your solution as a real number rounded to fourdecimal places or you may leave your solution in terms of the standard normal CDF, Φ.X is a discrete random variable such that the possible values are {a, a+1, a+2, ... b-1, b}. The random variable has a mean of 7 and a variance of 4. What is the Pr(X<=6|X>4)?Let X be a strictly positive random variable. Determine which of the following quantitiesare larger:(a) E(X^4) or (E(X))^4(b) E(X^1/9) or (E(X))^1/9(c) E(X^3 + 4X^2 + 9) or (E(X))^3 + 4(E(X))^2 + 9(d) E(e^X ) or e^E(X)(e) E(arctan(X)) or arctan(E(X))
- Find the variance of the random variable X with a PDF given by 3x2 S* if - 1< x< 1, f(x) = 2 otherwise. Submit your answer in fractional form.are iid random variables having mean u and variance o?. Form the random Suppose f1, 2, . sum SN .. E1 +..+ EN. (a) Derive the mean and variance of SN when N has a Poisson distribution with parameter d. (b) Determine the mean and variance of SN where N has a geometric distributions with mean (c) Compare the behaviors in (a) and (b) as A→ o.Let Z be nomrally distributed with mean 0 and variance 1. Let Z1 ,.., Zn be an iid random sample, all distributed as Z. Find the moment-generating function of Z2. please show all your work especially when intergrating. do not skip any minor details.
- Calculate the variance of a random variable X whose characteristic function is (1+ e3it)² f(t) = 4Let X be Bernoulli random variable, find the moment generating function X and find its mean, variance and third moment.1.) Consider the discrete random variable with pmf: P(X=x)=(A/(4^x)), x=2,3,4,5,.... What is the value of A?
