Find the expected value of the above random variable.
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- A diagnostic test for disease X correctly identifies the disease 88% of the time. False positives occur 11%. It is estimated that 2.78% of the population suffers from disease X. Suppose the test is applied to a random individual from the population. Compute the following probabilities. The percentage chance that, given a negative result, the person does not have disease X= The percentage chance that, the person will be misclassified =We toss three symmetrical coins. If all the coins are heads, we win $10, if all the coins are tails, we win $5, if exactly two heads are heads, we lose $10, and if exactly two tails are heads, we lose $1. Determine the distribution of the random variable that assigns a win to the result of the toss, calculate its expected value, variance, and standard deviation.A private hospital gets an average of 14 hospital admissions per week in its emergency room (ER). Let x be the outcome of hospital admissions in the emergency room in a week. In order to plan for the number of beds for the future, the hospital has to model the data with an appropriate distribution. (a) what distribution would you suggest for the random variable X, and why? (b) what is the probability of getting at most two admissions in one week? (c) what is the probability of getting at most two admissions in two weeks? (d) what is the probability of getting at least one admission in 2 days, assuming its ER opens 7 days a week? (e) if the hospital closes for three days because of a very serious event, how may new patients will the ER miss for admission on average, in those three days?
- Mike has a deck of standard poker cards with some cards missing. He wants to estimate the probability, p, of drawing a spade. Every day he makes an experiment. At the end of each experiment, he records the maximum likelihood estimator of the probability of observing a spade and places the drawn card back and shuffles the deck. Let X; be a random variable, taking value 1 if the i-th card draw of an experiment results in an even number and 0 if it results in an odd number. Clearly, X, ~ Bernoulli(p). Recall the MLE of a Bernoulli is p MLE The following are the outcomes of 5 experiments, one of which involved 8 card draws, two of which involved 6 card draws and two of which involved 4 card draws. Which experiments lead to the same MLE of p? Select all that apply. O (1 = 0, x2 = 1, r3 = 0, x4 = 0, r; 1, 6 = 0, x7 = 1, Is = 1, xg = 0, r10 = 1, 111 = 0, 112 = 0) O (1 = 1, a2 = 0, r3 = 1, x4 = 1) O ( = 0, x2 = 0, r3 = 0, r4 = 0) O (21 = 0, x2 = 1, £3 = 0, T4 = 0, x5 = 0, x6 = 0, r7 = 1, rg 0)…1. A bank operates a drive-up and walk-up window. Let X = the proportion of the time the drive-up facility is in use and let Y = the proportion of the time the walk-up window is in use. Suppose the operations manager uses the following distribution as the joint pdf based on his historical data: f(x, y) = 5 - 1/2(x + y²) if 0 (x + y²) if 0 ≤ x ≤ 1 and 0 ≤ y ≤1 otherwise (a) Find the marginal pdf for X, fx(x). (b) Find the marginal pdf for Y, fy(y).Show that E(p) =p where p = , the probability of success for a binomial random variable X. What does this say about the estimator p?
- 1A small business is studying the number of customers that enter the store. They record the number of customers entering every hour for three days, and find that 27% of the time two customers enter the store in an hour, 28% of the time of the time one customer enters the store in an hour, and the remainder of the time no one enters the store in an hour. What is the expected number of customers per hour?A fair coin is flipped 100 times. The random variable X= number of heads out of the 100 tosses. What is the expected value of X?
- Provide an example of a binomial random variable and explain how each condition for the binomial distribution is fulfilled.A fair coin is tossed three times (consecutively). If X is the number of heads of the first shot and Y is the random variables representing the total number of heads, what is the expected value of Y? calculate.
