QuestionLet be a non-negative random variable with a distribution such that for all . Calculate .
Studdy Solution
STEP 1
What is this asking? What's the probability that *e* to the power of a random variable minus 0.29 is less than or equal to 1, given how is distributed? Watch out! Don't forget to consider the properties of the exponential distribution and inequalities when solving for .
STEP 2
1. Rewrite the probability inequality.
2. Evaluate the probability.
STEP 3
We want to find .
Let's **isolate** by adding 0.29 to both sides of the inequality:
STEP 4
To **isolate** , we'll take the natural logarithm (ln) of both sides.
Since the natural logarithm is an increasing function, the inequality sign remains the same:
STEP 5
Now, let's **calculate** the value of : So, we're looking for .
STEP 6
We're given .
We want , which is the complement of .
Remember, the **total probability** of all possible outcomes is 1.
STEP 7
Since the total probability is 1, we can write:
STEP 8
Now, we can **substitute** into the given distribution:
STEP 9
Finally, we **substitute** this back into our complement equation:
STEP 10
The probability is approximately **0.1414**.
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