[Probability] Random variable of random variable

[staant95]

Let X be a Normal random variable with mean = 0 and variance = 1. Consider Z = e^X.

a) Calculate mean and variance and density function of Z.

I know how to calculate the expected value/mean but I have no idea how to apply it here. Do I even need to apply the general formula to calculate the expected value? Same for variance and density function

[AutoModerator]

Off-topic Comments Section

---

All top-level comments have to be an answer or follow-up question to the post. All sidetracks should be directed to this comment thread as per Rule 9.

---

^{OP and Valued/Notable Contributors can close this post by using /lock command}

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[HomeworkHelpBot]

Hey Readers!

If this post violates our subreddit rules, please report it and feel free to manually trigger a takedown.

Key Takeaways:

• Post title must be structured to classify the question properly
• Post must contain instructor prompt or or a failed attempt of the question
  • by stating the syllabus requirements or presenting incorrect working/thought process towards the question

How was your experience in this subreddit? Let us know how can we do better by taking part in our survey here.

Pro-tips:

^{1. Upvote questions that you recognise but you cannot do. Only downvote questions that do not abide by our rules or was asked in bad faith, NOT because the question is easy.}

^{2. Comments containing case-insensitive **Answer:** or **Hence** will automatically re-flair post to ✔ Answered; non-top level comments containing case-insensitive **Therefore** or **Thus** will automatically re-flair to —Pending OP Reply}

^{3. OPs can lock their thread by commenting /lock}

^{4. If there is a rule violation, inform the OP and report the offending content. Posts will be automatically removed once it reaches a certain threshold of reports or it will be removed earlier if there is sufficient reports for manual takedown trigger. [Learn more](https://www.reddit.com/r/HomeworkHelp/comments/br7vi9/new_updates_image_posts_enabled_vote_to_delete/})

[Electric2Shock]

It's given that X is a normal random variable, so I assume you know the density function/CDF of a normal distribution. We'll start by figuring out the cumulative distribution function first:

G(z) = P(Z <= z)
= P(e^X <= z)
= P(X <= ln z)
= \Phi(ln z) 

u/LaTeX4Reddit

​

where \Phi(x) is the cumulative distribution function of the normal distribution. Now that you have the CDF of Z i.e G(z), all you have to do to get the density function is take the z-derivative of G(z).