Suppose a continuous random variable X has the CDF(a) What is the range of possible outcomes for X?(b) As far as you can tell from the graph, what is P( X ≤ 1.5 )?(c) What feature of the CDF tells you that X is a continuous random variable?
Question
Suppose a continuous random variable X has the CDF(a) What is the range of possible outcomes for X?(b) As far as you can tell from the graph, what is P( X ≤ 1.5 )?(c) What feature of the CDF tells you that X is a continuous random variable?
Solution 1
(a) The range of possible outcomes for X is the set of all real numbers for which the cumulative distribution function (CDF) is defined and is between 0 and 1. In general, for a continuous random variable, this is typically the entire real line (from negative infinity to positive infinity), unless specified otherwise.
(b) Without a specific graph, it's impossible to determine the exact value of P(X ≤ 1.5). However, in general, P(X ≤ x) is given by the value of the CDF at x. So, P(X ≤ 1.5) would be the value of the CDF at 1.5.
(c) The feature of the CDF that tells you that X is a continuous random variable is that the function is continuous itself. For a continuous random variable, the CDF is a continuous, non-decreasing function. This is in contrast to a discrete random variable, where the CDF would be a step function, jumping at each value that the variable can take on.
Solution 2
I'm sorry, but I can't provide the help you're looking for because the graph of the Cumulative Distribution Function (CDF) is not provided.
However, I can guide you on how to answer these questions if you have the graph:
(a) The range of possible outcomes for X is typically given by the x-values for which the CDF is defined. You would look at the graph and identify the smallest and largest x-values for which the function has a value.
(b) To find P(X ≤ 1.5), you would look at the y-value of the CDF at x = 1.5. This is because the value of the CDF at a given x-value is the probability that X takes on a value less than or equal to that x-value.
(c) The feature of the CDF that tells you X is a continuous random variable is that the graph is a continuous curve without any jumps or gaps. For a discrete random variable, the CDF would have jumps where the probability mass function has positive probability.
Solution 3
(a) The range of possible outcomes for X is the set of all real numbers for which the cumulative distribution function (CDF) is greater than 0 and less than 1. In other words, it's the set of all x such that 0 < F(x) < 1. This is because the CDF of a random variable X is a function that gives the probability that X will take a value less than or equal to x.
(b) Without a specific graph, it's impossible to give a precise value for P(X ≤ 1.5). However, in general, P(X ≤ x) is given by the value of the CDF at x. So, P(X ≤ 1.5) would be F(1.5), where F is the CDF of X.
(c) The feature of the CDF that tells you X is a continuous random variable is that the CDF is a continuous function. This means that for any two numbers a and b with a < b, there are no jumps or discontinuities in the function between a and b. In other words, the graph of the CDF is an unbroken curve. This is in contrast to the CDF of a discrete random variable, which would have jumps at the values that the variable can take.
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