probabilitystatistics
Random variable $X$ has the following discrete distribution:
$f(x) = k/x$ for $x = 1, 2, 3$
$f(x) = 0$ otherwise
Find $k$ so that $f(x)$ is a legitimate probability mass function
What is $E(X)$, the expected value of X?
Calculate $V(X)$, the variance of $X$.
Let $X_1$, $X_2$, …, $X_{36}$ be a random sample selected from the distribution of X. Use the central limit theorem to approximate the probability that the sample mean is greater than 2.1 but less than 2.5.
If the discrete random variable $X$ takes integer values, then $$P(X > x)= P(X \ge x+1) = P(X \ge x+.5)$$ The continuity correction would use the third expression when using a continuous distribution as an approximation.
Ordinarily, the approximating continuous distribution would have positive probability in the interval $[x, x+.5].$ In that case using the continuity correction will give you a smaller approximated value.
Example: Suppose $X \sim \mathsf{Binom}(n = 64, p = 1/2)$ and you seek $P(X > 30).$ The exact value is $P(X > 30) = 1 - P(X \le 30) = 0.6460096.$
1 - pbinom(30, 64, .5)
## 0.6460096
If you use $P(X^\prime > 30) = 1 - P(X^\prime \le 30)$ as an approximation, where $X^\prime \sim \mathsf{Norm}(\mu = 32, \sigma=4),$ you will get $P(X > 300) \approx 0.6914625.$
1 - pnorm(30, 32, 4)
## 0.6914625
But if you use the continuity correction, you will use $P(X^\prime > 30.5) = 1 - P(X^\prime \le 30.5) = 0.6461698.$ Hence, your approximation will be $P(X > 30) \approx 0.6461698.$ This is smaller than the value 0.6914625 without the continuity correction. It is also closer to the exact binomial probability.
1 - pnorm(30.5, 32, 4)
## 0.6461698
Usually in textbook examples you can expect about two decimal places of accuracy from a continuity-corrected normal approximation to a binomial distribution. To four decimal places, the exact value in this example is 0.6460 and the continuity-corrected normal approximation is 0.6462. (Here we get three-place accuracy; approximations are often best when $p \approx 1/2.$)
The figure below shows relevant binomial probabilities (vertical bars) and the approximating normal density curve. Notice that be binomial probability $P(X = 31)$ is approximated by the area under the normal curve above the interval $[30.5, 31.5].$ The uncorrected approximation wrongly includes the vertical strip between $x = 30.0$ and $x=30.5$ under the normal curve.
Note: The values I have shown are from R statistical software. If your normal approximations are obtained by standardization and using a printed normal table, then results will be slightly different because of the rounding entailed in the use of the table.
Under certain conditions, the probability $P(\bar{X}_n \in [-1, 1])$ (for example), which is a non-random number, can be approximated by $P(Y \in [-1, 1])$ for some normal random variable $Y$ with appropriate mean and variance, using the central limit theorem. To emphasize, you are approximating the probability $P(\bar{X}_n \in [-1, 1])$, and not the random variable $\bar{X}_n$ itself: you are not saying something is close to $\bar{X}_n$.
Best Answer
Hints:
1) What is the definition of a probability mass function? What has to be true if order for the given $f$ to satisfy the definition?
2) What is the definition of the expected value of a discrete random variable? What do you get (after answering (1)) when you plug in the given $f$?
3) Similar to (2)
4) The answers to (2) and (3) give you all the information about $X$ that you need for this.