Let 𝑋∼𝖤𝗑𝗉(𝜆) for some 𝜆>0. Compute its first two moments:

Question

Let 𝑋∼𝖤𝗑𝗉(𝜆) for some 𝜆>0. Compute its first two moments:

Answer 1

The expected value of a random variable 𝑋 following an exponential distribution with parameter 𝜆 is given by:

𝔼[𝑋]=1/𝜆

To compute the second moment, we need to use the moment generating function of 𝑋:

𝑀(𝑡)=𝔼[e^(𝑡𝑋)]=∫_(0^(∞) ) e^(𝑡𝑥) 𝜆e^(−𝜆𝑥)𝑑𝑥

𝑀(𝑡)=𝜆∫_(0^(∞) ) e^((𝑡𝑥−𝜆𝑥))𝑑𝑥

𝑀(𝑡)=𝜆∫_(0^(∞) ) e^((𝑡−𝜆)𝑥)𝑑𝑥

Using integration by parts, we can evaluate the integral as:

𝑀(𝑡)=𝜆/((𝜆−𝑡))

Then, the second moment is given by:

𝔼[𝑋^2]=𝑀′′(0)=𝑑^2/d𝑡^2 𝑀(𝑡)|_(𝑡=0)

Differentiating 𝑀(𝑡) twice, we get:

𝑀′′(𝑡)=2𝜆^2/(𝜆−𝑡)^3

Evaluating at 𝑡=0, we obtain:

𝔼[𝑋^2]=2(1/𝜆)^2=2/𝜆^2

Therefore, the first two moments of 𝑋 are:

𝔼[𝑋]=1/𝜆

𝔼[𝑋^2]=2/𝜆^2

Answer 2

To compute the first two moments of a random variable 𝑋 that follows an exponential distribution with parameter 𝜆, we need to find the mean (first moment) and the variance (second moment).

1. Mean (First Moment):
The mean of an exponential distribution with parameter 𝜆 is given by 𝐸[𝑋] = 1/𝜆.
So, the mean of 𝑋 is 𝐸[𝑋] = 1/𝜆.

2. Variance (Second Moment):
The variance of an exponential distribution with parameter 𝜆 is given by 𝑉𝑎𝑟[𝑋] = 1/𝜆^2.
So, the variance of 𝑋 is 𝑉𝑎𝑟[𝑋] = 1/𝜆^2.

Therefore, the first two moments of the random variable 𝑋 that follows an exponential distribution with parameter 𝜆 are:
Mean (First Moment): 𝐸[𝑋] = 1/𝜆
Variance (Second Moment): 𝑉𝑎𝑟[𝑋] = 1/𝜆^2